Keywords: Stochastic Modeling and Stochastic Systems Theory
Abstract: The usual way to think about randomness is to start from the discrete world, where one tosses a finite collection of random variables. When the randomness is in fact “spread” in a fairly natural way over space, then in some cases, rather complex and interesting random geometric physically relevant structures with some surprising features can emerge. The aim of this talk is to illustrate this using some specific examples that arose in recent works.

Keywords: Optimal Control, Feedback Control Systems, Robotics
Abstract: We propose a framework for quantitatively reasoning about control architectures that highlights the importance of layered control architectures. Our approach takes an overall synthesis problem and decomposes it into tractable subproblems via suitable relaxations and approximations, with each subproblem assigned to a layer.

Keywords: Optimal Control, Optimization : Theory and Algorithms, Feedback Control Systems
Abstract: For optimal control problems that involve planning and following a trajectory, trajectory generators combined with two degree of freedom (2DOF) controllers are a ubiquitously used control architecture that decomposes the problem into a trajectory generation layer and a feedback control layer. However, despite the broad use and practical success of this layered control architecture, it remains a design choice that must be imposed emph{a priori} on the control policy. To address this gap, this paper seeks to initiate a principled study of the design of layered control architectures, with an initial focus on the 2DOF controller. We show that applying the Alternating Direction Method of Multipliers (ADMM) algorithm to solve a strategically rewritten optimal control problem results in solutions that are naturally layered, and composed of a trajectory generation layer and a feedback control layer. Furthermore, these layers are coupled via Lagrange multipliers that ensure dynamic feasibility of the planned trajectory. We instantiate this framework in the context of deterministic and stochastic linear optimal control problems, and show how our approach automatically yields a feedforward/feedback-based control policy that exactly solves the original problem.

Keywords: Artificial Intelligence, Robotics, Model Predictive Control
Abstract: The impressive capabilities of Large Language Models (LLMs) have led to various efforts to perform robotic control via natural language commands. The goal is for the motor- control task to be performed accurately, efficiently and safely, while also enjoying the flexibility imparted by LLMs to design and adjust the task through natural language. In this work, we demonstrate how a careful layering of an LLM in combination with a Model Predictive Control (MPC) (Rawlings et al. (2017)) formulation allows for accurate, safe, and flexible robotic control via natural language. As a proof of concept, we conduct extensive experiments in simulation and hardware for a robotic, demonstrating that our proposed architecture successfully executes complex tasks while adhering to safety constraints. The approach presented can be used for different platforms by simply modifying the low-level controller.

Keywords: Machine Learning and Control, Optimal Control, System Identification
Abstract: We consider two standard learning problems in control: identifying a model from data, i.e., system identification, and model-free control via policy gradient methods. Both problems are well studied and in the LTI and LQR problem setups have well known sample complexity bounds. In this talk we consider a collaborative learning setup where we have two conflicting goals: agents are allowed to collaborate by sharing models learnt from their local data with the objective of producing an aggregate model that performs well for each client. However, we would also like the models to allow for downstream personalization. Thus we need to generalize with an eye on personalization. This talk will highlight various collaboration and communication scenarios for which we have been able to pinpoint the sample complexity benefits/hindrances achieved through collaboration.

Keywords: Large Scale Systems, Computations in Systems Theory, Optimization : Theory and Algorithms
Abstract: When designing autonomous systems, we need to consider multiple trade-offs at various abstraction levels, and choices of single (hardware and software) components need to be studied jointly. For instance, the design of future mobility solutions (e.g., autonomous vehicles) and the design of the mobility systems they enable are closely coupled. Indeed, knowledge about the intended service of novel mobility solutions would impact their design and deployment process, whilst insights about their technological development could significantly affect transportation policies.

Co-designing autonomous systems is a complex task for at least two reasons. First, the co-design of interconnected systems (e.g., networks of cyber-physical systems) involves the simultaneous choice of components arising from heterogeneous fields, while satisfying systemic constraints and accounting for multiple objectives. Second, components are connected via interactions between different stakeholders. I will present a framework to co-design such systems, leveraging a monotone theory of co-design. The framework will be instantiated in applications in mobility and autonomy. Through various case studies, I will show how the proposed approaches allow one to efficiently answer heterogeneous questions, unifying different modeling techniques and promoting interdisciplinarity, modularity, and compositionality. I will then discuss open challenges for compositional systems design optimization.

Keywords: Coding Theory
Abstract: In this talk we present a way of constructing spreads of Fq^n. They will arise as orbits under the action of an Abelian non-cyclic group. First we construct a family of orbit codes of maximum distance using this group and then we complete each of these codes to achieve a spread of the whole space having an orbital structure.

Keywords: Coding Theory
Abstract: We are going to consider a new construction of convolutional codes and analyze their erasure correction capability. MDP convolutional codes have optimal correction capacity if decoding is done sequentially. However, there are not many constructions of MDP convolutional codes and the existing ones require a large finite field. We will construct new codes by considering an encoder of an MDP convolutional code and repeating a (matrix) coefficient of the encoder. The resulting code, which we call Pseudo-MDP, will be defined over the same field. Although it may not be MDP, it will have a better erasure correction capacity than the original code.

Keywords: Coding Theory
Abstract: This work provides a continuation of earlier work on simplex distributed storage block codes over extension fields, published at MTNS 2014. We first look at easy repair properties of unit memory tailbiting convolutional codes. We next construct a convolutional code with simplex properties. We explore its easy repair properties for the distributed storage setting, as well as the usage of a sliding window decoder to do the actual easy repair.

Keywords: Coding Theory
Abstract: A distributed storage system stores data over multiple storage nodes. The main goal of an exact repair scheme is to recover the data from a failed node by accessing and downloading information from the rest of the nodes. In a groundbreaking paper, Guruswami and Wootters (2017) developed an exact repair scheme using Reed Solomon codes. In these notes, we extend the repair scheme to the family of Reed-Muller codes.

Keywords: Coding Theory, Information Theory, Optimization : Theory and Algorithms
Abstract: This paper provides a semidefinite programming hierarchy based on state polynomial optimization to determine the existence of quantum codes with given parameters. The hierarchy is complete, in the sense that if a (!(n,K,delta)!)_2 code does not exist, some level of the hierarchy is infeasible. It is not restricted to stabilizer codes and thus applicable generally. While it is formally dimension-free, we restrict it to qubit codes by making use of quasi-Clifford algebras. From an intermediate level, a quantum Lov'asz bound for self-dual quantum codes is derived. A symmetry reduction of a minor variation of this Lov'asz bound then recovers the quantum Delsarte bound. A second symmetry reduction making use of the Terwilliger algebra leads to semidefinite programming bounds of size O(n^4).

Keywords: Optimization : Theory and Algorithms, Infinite Dimensional Systems Theory

Keywords: Optimization : Theory and Algorithms, Information Theory
Abstract: We study the set of correlations that can be achieved in quantum theory in the simplest Bell scenario---with two inputs and outputs per party---where qubits suffice to obtain extremal correlations. This qubit reduction allows us to cast the problem of maximising a Bell inequality as a polynomial optimisation problem. Using results from polynomial optimisation, we show that for every Bell inequality there exists a finite semidefinite programme that finds its maximum. Furthermore, using a connection to entanglement theory, we show that there exists a single semidefinite programme that finds the maximum of an arbitrary Bell inequality. We further discuss the (distinct) question of the semidefinite representability of the set of quantum correlations in the simplest Bell scenario.

Keywords: Optimization : Theory and Algorithms, Numerical and Symbolic Computations
Abstract: Using standard tools of harmonic analysis, we state and solve the problem of moments for positive measures supported on the unit ball of a Sobolev space of multivariate periodic trigonometric functions. We describe outer and inner semidefinite approximations of the cone of Sobolev moments. They are the basic components of an infinite-dimensional moment-sums of squares hierarchy, allowing to solve numerically non-convex polynomial optimization problems on infinite-dimensional Sobolev spaces, with global convergence guarantees.

Keywords: Large Scale Systems, Machine Learning and Control, Optimization : Theory and Algorithms
Abstract: Optimal transport (OT) has gained widespread recognition across various fields, from economics and fluid mechanics, lately, to machine learning. However, its connection with dynamical systems and control and potential to drive applications in these areas has not been fully realized. To fill this gap, we propose an OT-based ensemble control framework, which enables the modeling of the OT process as an ensemble control system and, conversely, the solution to ensemble control problems using OT techniques. Our development is rooted in the concept of treating OT as a time-dependent process by utilizing displacement interpolation. Through this interpretation, we develop a moment kernelization approach to model the OT process as an ensemble system using moment representations. This framework naturally gives rise to a systematic representation learning approach to modeling OT with control systems.

Keywords: Optimal Control, Machine Learning and Control, Neural Networks
Abstract: In this study, we introduce a novel method for generating i.i.d. synthetic samples from high-dimensional real-valued probability distributions, defined by Ground Truth (GT) samples. Our approach integrates space-time mixing strategies across temporal and spatial dimensions. The methodology involves three interrelated stochastic processes: (a) linear processes with space-time mixing yielding Gaussian conditional densities, (b) their diffusion bridge analogs conditioned on initial and final states, and (c) nonlinear stochastic processes refined via score-matching techniques. The training regime fine-tunes both nonlinear and potentially linear models to align with GT data. We validate our space-time diffusion bridge approach through numerical experiments.

Keywords: Optimization : Theory and Algorithms
Abstract: Gromov–Wasserstein (GW) transport allows for the matching of objects based on the preservation of their internal geometry, which does not require an embedding in a canonical ambient space. Hence, GW barycenters are perfectly suited for the joint embedding-free matching and interpolation of multiple objects. However, the high computational costs of existing algorithms still makes it challenging when considering larger inputs such as three-dimensional surfaces. In this short note, we consider GW barycenters and their relation to multi-marginal GW transport. These multi-marginal formulations spark two numerical approaches for computing these barycenters which we briefly discuss.

Keywords: Optimization : Theory and Algorithms, Large Scale Systems, Stochastic Control and Estimation
Abstract: In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problems. This type of problems can be seen as a generalization of multi-marginal optimal transport problems with graph-structured costs, and appear for instance in unbalanced optimal transport, and nonlinear density control problems, such as multi-species potential mean field games. We develop an efficient algorithm for solving these types of problems, and showcase the flexibility of our method on a density control problem.

Keywords: Optimal Control, Optimization : Theory and Algorithms, Quantum Control
Abstract: We address the problem of symmetry reduction of optimal control problems under the action of a finite group from a measure relaxation viewpoint. We propose a method based on the moment-SOS aka Lasserre hierarchy which allows one to significantly reduce the computation time and memory requirements compared to the case without symmetry reduction. We show that the recovery of optimal trajectories boils down to solving a symmetric parametric polynomial system. Then we illustrate our method on the time-optimal inversion of qubits.

Keywords: Optimization : Theory and Algorithms
Abstract: The Positivstellensätze of Putinar and Schmüdgen show that any polynomial f positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellensätze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this work, we restrict to the fundamental case of the hypercube. We show an upper degree bound for Putinar-type representations on the cube of the order O(fmax/fmin), where fmax, fmin are the maximum and minimum of f on the cube, respectively. Previously, specialized results of this kind were available only for Schmüdgen-type representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound. This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities.

Keywords: Operator Theoretic Methods in Systems Theory, Numerical and Symbolic Computations
Abstract: We study the connections between operator moment sequences {mathcal T}=displaystyle(T_n)_{ninmathbf{N}} of self-adjoint operators on a complex Hilbert space mathcal{H} and the local moment sequences langle{mathcal T}x,xrangle = (langle T_nx,xrangle)_{ninmathbf{N}} for arbitrary xin mathcal{H}. We provide necessary and sufficient conditions for solving the operator moment problem on mathbf{R}, and we show that these criteria are automatically valid on compact subsets of mathbf{R}. Applications of the compact case are used to study subnormal operator weighted shifts. A Stampfli-type propagation theorem for subnormal operator weighted shifts is also established. In addition, we discuss the validity of Tchakaloff's Theorem for operator moment sequences with compact support. In the case of a recursively generated sequence of self-adjoint operators, necessary and sufficient conditions for an affirmative answer to the operator recursive moment problem are provided, and the support of the associated representing operator-valued measure is described.

Keywords: Optimization : Theory and Algorithms, Applications of Algebraic and Differential Geometry in Systems Theory
Abstract: Let Cscr be a compact subset of RR^{n} and f: Cscr to RR be a emph{real analytic} function assumed to be a Morse function. We consider the problem of computing emph{all} local minimizers. Note that, under our assumptions, there are finitely many such points.

We use the classical model of Turing machines. Since, not all real numbers can be represented in this model (an infinite amount of bits may required to represent them), we assume that the function f is given by an emph{evaluation program} Gamma which takes as input rational points in Cscrcap QQ^{n} considering both the framework where this evaluation program is emph{exact} -- if the image of f can be represented with a finite amount of bits -- or emph{noisy} -- in that case, we assume that the evaluation function takes an additional parameter eta and returns a rational point at distance at most eta to the real point it should compute in the image of f.

Under these assumptions, we design an algorithm which takes as input Gamma (and eta in the noisy model) and a numerical accuracy parameter epsilon. With some additional assumptions of probabilistic nature, it outputs finitely many rational points of Cscrcap QQ^{n} such that union of the balls of radius varepsilon centered at these points contains the set of all local minimizers of the function f.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Optimization : Theory and Algorithms
Abstract: We give bounds on the geodesic distances on the Stiefel manifold, derived from new geometric insights. These geodesic distances are induced by a previously proposed one-parameter family of Riemannian metrics, which contains the well-known Euclidean and canonical metrics. We give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the Frobenius distance. These bounds aim at improving the performance of minimal geodesic computation algorithms and contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their practical applications.

Keywords: Numerical and Symbolic Computations, Optimal Control

Keywords: Optimization : Theory and Algorithms, Applications of Algebraic and Differential Geometry in Systems Theory, Machine Learning and Control
Abstract: In this work, we state a general conjecture on the solvability of optimization problems via algorithms with linear convergence guarantees. We make a first step towards examining its correctness by fully characterizing the problems that are solvable via Riemannian gradient descent with linear convergence.

Keywords: Optimization : Theory and Algorithms
Abstract: On a manifold or a closed subset of a Euclidean vector space, a retraction enables to move in the direction of a tangent vector while staying on the set. Retractions are a versatile tool to perform computational tasks such as optimization, interpolation, and numerical integration. This paper studies two known definitions of retraction on a closed subset of a Euclidean vector space, one being weaker than the other. Specifically, it shows that, in the context of constrained optimization, the weaker definition should be preferred as it inherits the main property of the other while being less restrictive.

Keywords: Model Predictive Control, Machine Learning and Control
Abstract: Reinforcement Learning (RL) is a very successful data-driven approach to optimal control, which, however, typically struggles to provide both explainability and strong guarantees on the behavior of the resulting control scheme, e.g., safety and stability. In contrast, Model Predictive Control (MPC) is a well-established tool for the closed-loop optimal control of complex systems subject to constraints. MPC benefits from a rich theory that allows one to provide strong guarantees about closed-loop behavior. Because of model inaccuracy, however, MPC can fail to deliver satisfactory closed-loop performance. The use of MPC as a function approximator within RL paves the way toward safe and explainable RL but also requires the development of a new theory. In this paper, we will mention some recent theoretical results that make it possible to introduce safety and stability guarantees in RL by using MPC as function approximator. Furthermore, we will discuss further open questions that are the subject of ongoing research.

Keywords: Optimal Control, Optimization : Theory and Algorithms, Machine Learning and Control
Abstract: Optimal actuator and control design is studied as a multi-level optimisation problem, where the actuator design is evaluated based on the performance of the associated optimal closed loop. The evaluation of the optimal closed loop for a given actuator realisation is a computationally demanding task, for which the use of a neural network surrogate is proposed. The use of neural network surrogates to replace the lower level of the optimisation hierarchy enables the use of fast gradient-based and gradient-free consensus-based optimisation methods to determine the optimal actuator design. The effectiveness of the proposed surrogate models and optimisation methods is assessed in a test related to optimal actuator location for heat control.

Keywords: Stochastic Control and Estimation, Dissipativity
Abstract: Analyzing optimal control problems in deterministic settings has benefitted from the concepts of turnpike and dissipativity. We extend the classic dissipativity notion of Jan C. Willems to stochastic systems, introducing two distinct dissipativity notions based on stationarity concepts in distribution and random variables. We explain the links between these notions and explore their connections to various forms of stochastic turnpike properties. The proposed turnpike properties range from a formulation for random variables via turnpike phenomena in probability and in probability measures to a turnpike property for single moments. Finally, we introduce the generalized linear-quadratic stochastic optimal control problem as an example for which an explicit storage function can be constructed and illustrate our analytical findings by numerical simulations.

Keywords: Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory, Robotics
Abstract: Robust autonomy stacks require tight integration of perception, motion planning, and control layers, but these layers often inadequately incorporate inherent perception and prediction uncertainties. Robots with nonlinear dynamics and complex sensing modalities operating in an uncertain environment demand more careful consideration of how uncertainties propagate across stack layers. We propose a framework to integrate perception, motion planning, and control by explicitly incorporating perception and prediction uncertainties into planning so that risks of constraint violation can be mitigated. Specifically, we use a nonlinear model predictive control based steering law coupled with a decorrelation scheme based Unscented Kalman Filter for state and environment estimation to propagate the robot state and environment uncertainties. Subsequently, we use Distributionally Robust (DR) risk constraints to limit the risk in the presence of these uncertainties. Finally, we present a layered autonomy stack consisting of a nonlinear steering-based DR motion planning module and a reference trajectory tracking module. Our numerical experiments with nonlinear robot models and an urban driving simulator show the effectiveness of our proposed approaches.

Keywords: Stochastic Control and Estimation, Transportation Systems, Stochastic Modeling and Stochastic Systems Theory
Abstract: We consider the optimal control of the inflow into a supply system given an uncertain demand stream. Thereby, the supply system is modeled by a hyperbolic partial differential equation (PDE), more precisely a hyperbolic balance law. A stochastic differential equation is used to describe the uncertain demand. We add chance constraints to enhance supply reliability. We will present the generic procedure from Göttlich et al. (2021) how to come up with a deterministic reformulation of this stochastic optimal control problem. We use well-established deterministic PDE-constrained optimization techniques. We will illustrate our procedure for an energy supply control problem given an uncertain consumer demand.

Keywords: Nonlinear Filtering and Estimation

Keywords: Physical Systems Theory

Keywords: Infinite Dimensional Systems Theory, Optimization : Theory and Algorithms

Keywords: Coding Theory, Information Theory
Abstract: Let {mathbb F}_q be a finite field with q equiv 3 ; {rm mod} ; 4. Let w be a primitive element of {mathbb F}_{q^4}^* and let mathcal{C}(w) be the irreducible cyclic code of length (q^4-1)/(q-1) of dimension 4 defined by w. The symbol-pair weight enumerator of mathcal{C}(w) is determined exactly by the invariant mu(w) introduced in cite{ZSO}. The determination of good upper and lower bounds on this invariant remains an open problem. Let W be the set of all primitive elements of {mathbb F}_{q^4}^*. In this paper, by using algebraic and arithmetical methods, particularly those from the theory of algebraic curves over finite fields, we derive effective upper and lower bounds on sum_{w in W} mu(w).

Keywords: Coding Theory, Information Theory
Abstract: There exist in the literature two natural generalizations of low density parity check (LDPC) codes: 1) LDPC convolutional codes or sometimes also called spatially coupled LDPC codes which have shown to be able to reach Shannon capacity in a practical way. 2) Moderate density parity check (MDPC) codes which are linear codes possessing a parity check matrix whose row weight is not more than O(√n), still allowing efficient decoding with high probability. MDPC codes became a highly interesting class of block codes for the purpose of doing code- based cryptography. In this paper we study MDPC convolutional codes and some of their basic properties. At the end of the paper we explain how one can construct a new code-based cryptographic system. This new system can be seen as a convolutional version of the famous BIKE system currently evaluated for possible standardization by the National Institute for Standards and Technology (NIST).

Keywords: Coding Theory, Information Theory, Mathematical Theory of Networks and Circuits
Abstract: The aim of this note is to study under which conditions a family of constant dimension codes produces flag codes. In that situation, we realise that diferent flag codes can come from the same family of constant dimension codes. Then we give a method to obtain the largest flag code generated by a family of constant dimension codes and we determine in which cases such a family gives exactly one flag code. These notions will be crucial to properly connect the (semi)linear equivalence of flag codes with the (semi)linear equivalence of their projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes.

Keywords: Optimization : Theory and Algorithms, Numerical and Symbolic Computations, Information Theory

Keywords: Optimization : Theory and Algorithms, Numerical and Symbolic Computations, Operator Theoretic Methods in Systems Theory
Abstract: The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for the local Hamiltonian problem in quantum complexity theory. In this talk we attack this problem using the algebraic structure of QMC, specifically the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. We give a new hierarchy of semidefinite programming (SDP) relaxations to QMC by extending non-commutative Sum of Squares optimization techniques. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by qubit swap operators. We also give a polynomial-time algorithm that exactly computes the maximum eigenvalue of the QMC Hamiltonian for graphs that can be “decomposed” as a signed combination of cliques.

Keywords: Optimization : Theory and Algorithms, Quantum Control, Information Theory
Abstract: We present a connection between two very different problems: the joint measurability of measurements in quantum information theory and the inclusion problem for free spectrahedra. In particular, joint measurability of a tuple of dichotomic quantum measurements is equivalent to the inclusion of the matrix diamond inside a free spectrahedron defined by the measurements. The same correspondence holds for general quantum measurements and a different free spectrahedron, the matrix jewel. Moreover, the amount of noise necessary to render any tuple of measurements compatible corresponds to the inclusion constants for the respective free spectrahedron. In order to compute the inclusion constants of interest and thereby the noise robustness of measurement incompatibility, we put forward hierarchies of semidefinite programs and demonstrate their usefulness with some examples.

Keywords: Optimization : Theory and Algorithms, Mechanical Systems
Abstract: The microstructure of metals and foams is often modelled using generalised Voronoi diagrams (Laguerre diagrams or anisotropic power diagrams). The challenge is to generate realistic geometric models with prescribed statistical properties, such as the distribution of the volumes and shapes of the cells. In this work we develop an efficient algorithm for generating anisotropic power diagrams with cells of prescribed volumes. Our approach uses semi-discrete optimal transport theory with an anisotropic cost, combined with a fast GPU implementation using the KeOps library.

Keywords: Optimization : Theory and Algorithms, Stochastic Modeling and Stochastic Systems Theory, Stochastic Control and Estimation
Abstract: Many problems, from distributionally robust optimization to learning biological dynamics, can be cast as optimization problems in the probability space, whereby the decision variable is a probability measure. In this talk, we review some of these problems and present an optimization framework to attack them. Our tools are enabled by the theory of optimal transport and Wasserstein gradient flows, together with more classical calculus of variations. In particular, we present first-order necessary and sufficient conditions for optimality, and we make sense of expressions like ”set the (Wasserstein) gradient to zero” or ”(Wasserstein) gradients are aligned optimality”, widely used for optimization in Euclidean settings. We then demonstrate that these simple and interpretable optimality conditions can be directly leveraged to study many optimization problems in the probability space, both to derive quasi-closed-form solutions (e.g., in a distributionally robust control problem) and to devise numerical algorithms (e.g., in learning population dynamics). This extended abstract is based on our papers (Lanzetti, Bolognani, and Dörfler, 2022; Lanzetti, Terpin, and Dörfler, 2024).

Keywords: Signal Processing, Optimization : Theory and Algorithms, Nonlinear Filtering and Estimation
Abstract: In this work, we consider modeling multi-sensor broad-band signals by means of the concept of a spatio-temporal spectrum, defined on the product space of spatial and frequency domains. In particular, we propose to track the evolution of time-vaying spatio-temporal spectra by means of a group-sparse optimal transport formulation. As we show, this allows us to fuse information across both separate time-instances and across frequency, leading to accurate estimates of the frequency content and location of broad-band signal sources

Keywords: Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory, Optimal Control
Abstract: The deterministic variant of the Lambert's problem was posed by Lambert in the 18th century and its solution for conic trajectory has been derived by many, including Euler, Lambert, Lagrange, Laplace, Gauss and Legendre. The solution amounts to designing velocity control for steering a spacecraft from a given initial to a given terminal position subject to gravitational potential and flight time constraints. In recent years, a probabilistic variant of the Lambert's problem has received attention in the aerospace community where the endpoint position constraints are softened to endpoint joint probability distributions over the respective positions. Such probabilistic specifications account for the estimation errors, modeling uncertainties, etc. Building on a deterministic optimal control reformulation via analytical mechanics, we show that the probabilistic Lambert's problem is a generalized dynamic optimal mass transport problem where the gravitational potential plays the role of an additive state cost. This allows us to rigorously prove the existence-uniqueness of the solution for the probabilistic Lambert problem both with and without process noise. In the latter case, the problem and its solution correspond to a generalized Schrödinger bridge, much like how classical Schrödinger bridge can be seen as stochastic regularization of the optimal mass transport. We deduce the large deviation principle enjoyed by the Lambertian Schrödinger bridge. Leveraging these newfound connections, we design a computational algorithm to illustrate the nonparametric numerical solution of the probabilistic Lambert's problem.

Keywords: Optimization : Theory and Algorithms, Coding Theory
Abstract: We compute the second level of the Lasserre hierarchy for the equiangular lines problem with a fixed angle. We show how the resulting semidefinite programs can be analyzed asymptotically in the dimension, and use this to give new linear bounds on the maximum number of equiangular lines with common angle arccos(alpha). We then extend these techniques to compute the second level of the hierarchy for the kissing number problem, and use this to prove that the optimal kissing configuration in 4 dimensions (the D_4 root system) is unique.

Keywords: Nonlinear Systems and Control, Optimization : Theory and Algorithms, Infinite Dimensional Systems Theory
Abstract: Polynomial optimization has emerged as powerful tools for the analysis and control of ordinary differential equations. Inspired by this success, extensions to partial differential equations and variational problems have recently been sought. This extended abstract reviews some of these extensions, pointing out similarities and differences between various approaches. Remaining open questions are also identified.

Keywords: Optimization : Theory and Algorithms, Numerical and Symbolic Computations
Abstract: Studying subcones of the convex cone of positive semidefinite, real forms in a given number of variables and a given degree is an important topic in Real Algebraic Geometry. The sums of squares cone (SOS) on the one hand, and the sums of nonnegative circuit forms (SONC) on the other hand are two subcones, which are studied extensively in the literature. In addition, the SOS+SONC cone consists of forms that decompose into a sum of an SOS and a SONC form. In this work, we highlight the set theoretic relation between two of the cones, respectively, and present different methods for showing that a given form is not contained in one of the cones.

Keywords: Optimization : Theory and Algorithms, Operator Theoretic Methods in Systems Theory
Abstract: This work focuses on minimizing the eigenvalue of a noncommutative polynomial subject to a finite number of noncommutative polynomial inequality constraints. Based on the Helton-McCullough Positivstellensatz, the noncommutative analog of Lasserre's moment-sum of squares hierarchy provides a sequence of lower bounds converging to the minimal eigenvalue, under mild assumptions on the constraint set. Each lower bound can be obtained by solving a semidefinite program. We derive complementary converging hierarchies of upper bounds. They are noncommutative analogues of the upper bound hierarchies due to Lasserre for minimizing polynomials over compact sets. Each upper bound can be obtained by solving a generalized eigenvalue problem.

Keywords: Optimization : Theory and Algorithms

Keywords: Optimization : Theory and Algorithms, Computations in Systems Theory, Numerical and Symbolic Computations
Abstract: In our earlier work Mlinarić et al. (2023), we showed that the interpolation-based iterative rational Krylov algorithm (IRKA) for H₂-optimal model reduction can be interpreted as a Riemannian gradient descent method with a fixed step size. Additionally, we found that the Petrov-Galerkin projection framework of IRKA can also be applied with variable step sizes. Transfer function IRKA (TF-IRKA) is a realization-independent, data-driven formulation of IRKA allowing to perform H₂-optimal approximation without needing a state-space formulation and projection. Here we present the equivalent Riemannian gradient descent formulation and interpretation of TF-IRKA and develop on the Loewner framework to support variable step sizes. We show that Riemannian gradient descent can be implemented using only transfer function samples as in the original TF-IRKA. The new Riemannian formulation with step size control guarantees stability of the reduced model. Numerical examples also show that smaller step sizes can mitigate some issues that may arise due to the ill-conditioning of Loewner matrices.

Keywords: Optimization : Theory and Algorithms, Signal Processing, Machine Learning and Control
Abstract: We study a type of Riemannian gradient descent (RGD) algorithm, designed through Riemannian preconditioning, for optimization on mathcal{M}_k^{mtimes n}---the set of mtimes n real matrices with a fixed rank k. Our analysis is based on a quotient geometric view of mrkk: by identifying this set with the quotient manifold of a two-term product space mathbb{R}^{mtimes k}timesmathbb{R}^{ntimes k} (matrices with full column ranks), we find an explicit form for the update rule of the RGD algorithm, which leads to a novel approach to analysing their convergence behavior in rank-constrained optimization. We then deduce some interesting properties that reflect how RGD distinguishes from other matrix factorization algorithms such as those based on the Euclidean geometry. We further show that this RGD algorithm is guaranteed to solve matrix sensing and matrix completion problems with linear convergence rate under the restricted positive definiteness property. Numerical experiments on matrix sensing and completion are provided to demonstrate these properties.

Keywords: Robotics, Optimization : Theory and Algorithms, Machine Learning and Control
Abstract: Riemannian methods are promising approaches to deal with rotations in robotic applications. In this work, we frame the grasping problem as a Bayesian inference of the posterior distribution given a success metric and an observation of the scene. We determine the grasp pose, which consists in the position and the orientation, by computing the maximum a posteriori with Riemannian gradient descent. While most previous works use Euler angles to represent the rotation part, our work focus on quaternions. We demonstrate the effectiveness of Riemannian optimization by illustrating that optimized trajectories pass through singularities when transformed into Euler angles and by performing a complex robotic grasping task with a high success rate.

Keywords: Infinite Dimensional Systems Theory, Optimal Control, Stability
Abstract: In this paper, we extend a classical approach to linear quadratic (LQ) optimal control via Popov operators to abstract linear differential-algebraic equations (ADAEs) in Hilbert spaces. To ensure existence of solutions, we assume that the underlying differential-algebraic equation has index one in the pseudo-resolvent sense. This leads to the existence of a degenerate semigroup that can be used to define a Popov operator for our system. It is shown that under a suitable coercivity assumption for the Popov operator the optimal costs can be described by a bounded Riccati operator and that the optimal control input is of feedback form. Furthermore, we characterize exponential stability of ADAEs which is required to solve the infinite horizon LQ problem.

Keywords: Infinite Dimensional Systems Theory, Port-Hamiltonian Systems, Linear Systems
Abstract: The existence of a Weierstraß form and the solvability for infinite dimensional differential algebraic equations with a radiality index is studied. In the latter, one makes use of integrated semigroups to determine a subset on which solutions exist and are unique. This gained information is later used for a special case of systems, namely abstract dissipative Hamiltonian differential algebraic equations.

Keywords: Optimal Control, Infinite Dimensional Systems Theory, Optimization : Theory and Algorithms
Abstract: This talk addresses the linear-quadratic optimal control problem for a class of partial differential-algebraic equations. The main objective is to solve an optimization problem on a finite-time horizon. Using calculus of variations, we prove the existence of a unique optimal control that minimizes a quadratic criterion. The optimal control is shown to be in a feedback form. Additionally, a system consisting of a differential Riccati-like equation coupled with an algebraic equation is derived. This system yields the solution to the optimal control problem in feedback form. Numerical simulations are conducted to illustrate application of the theory.