Keywords: Large Scale Systems, Optimal Control, Transportation Systems
Abstract: The state-feedback solution of the Linear Quadratic Regulator problem is central to control theory, in both education and research, as well as in practice. Despite its many successes, a weakness of the LQ Regulator is that input-output map of the optimal controller is almost always dense, even when the problem data is sparse. This makes the control difficult to implement in large-scale applications, due to the resulting expensive and complex communication requirements. In this work we demonstrate by example that while this input-output map is dense, the same control law may be efficiently computed using sparsity exploiting tools from linear algebra. This opens the possibility for the sparse implementation of optimal LQ Regulators for some simple applications.

Keywords: Large Scale Systems, Port-Hamiltonian Systems, Mathematical Theory of Networks and Circuits
Abstract: In this paper we consider model reduction by clustering, and show how this leads to spatial layered network and energy storage structures. Furthermore, we consider the Dirac structures that arise from multi-scale power-conserving interconnections in physical network systems and boundary control distributed-parameter systems.

Keywords: Linear Systems
Abstract: Bounding or diminishing the number of sign changes (variation) or local extrema in a signal is an intrinsic system property in, e.g., low-pass filtering or the over- and undershooting behaviour in the step-response of controlled systems. This work shows how to verify these properties for systems operators of linear time-invariant systems. In particular, by bounding the variation of input and output by an integer k, we will establish that variation bounding provides intermediates between externally positive systems (k = 0) and systems that are composed of simple first order lags (k = infty). Our results are exemplified for the observability/controllability operator, in which case we can provide new bounds for the number of sign changes in an impulse response and, thus, the number of local extrema in the step response.

Keywords: Linear Systems, Networked Control Systems, Optimal Control
Abstract: An optimal control problem on finite-dimensional positive cones is stated. Under a critical assumption on the cone, the corresponding Bellman equation is satisfied by a linear function, which can be computed by convex optimization. A separate theorem relates the assumption to existence of certain minimal elements in the dual cone. Two special cases are derived as examples. The first one, where the positive cone is a set to positive semi-definite matrices, reduces to standard linear quadratic control. The second one, where the positive cone is a polyhedron, reduces to a recent result on optimal control of positive systems.

Keywords: Nonlinear Systems and Control, Feedback Control Systems
Abstract: We propose and analyze the suitability of a spiking controller to engineer the locomotion of a soft robotic crawler. Inspired by the FitzHugh-Nagumo model of neural excitability, we design a bistable controller with an electrical flipflop circuit representation capable of generating spikes on-demand when coupled to the passive crawler mechanics. A proprioceptive sensory signal from the crawler mechanics turns bistability of the controller into a rhythmic spiking. The output voltage, in turn, activates the crawler’s actuators to generate movement through peristaltic waves. We show through geometric analysis that this control strategy achieves endogenous crawling. The electro-mechanical sensorimotor interconnection provides embodied negative feedback regulation, facilitating minimal controller tuning for locomotion. Dimensional analysis provides insights on the characteristic scales in the crawler’s mechanical and electrical dynamics, and how they determine the crawling gait. Adaptive control of the electrical scales to optimally match the mechanical scales can be envisioned to achieve further efficiency, as in homeostatic regulation of neuronal circuits. Our approach can scale up to multiple sensorimotor loops inspired by biological central pattern generators.

Keywords: Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory, Hybrid Systems
Abstract: In this paper we study the stochastic safety problem for a class of Markov chains generated by iterated function systems (IFS). These Markov chains represent a mathematical construct that describes the stochastic process of randomly iterating through a set of functions within the IFS. We propose a safety approach, which allows for the characterization of the reachability problem of the IFS through the lens of Markov chains.

Keywords: Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory, Systems on Graphs
Abstract: We discuss a problem introduced by Erwin Schröndiger in 1931/1932, to reconcile a prior probability law with data on marginal distributions at two points in time. Schrödinger’sinsight was to seek the most likely “posterior law” that is consistent with the given marginals. Later known as Schrödinger’s Bridge problem, the eponym problem bestowed inspiration on the theory of large deviations and various topics in uncertainty control. However, a compelling unstudied facet of this program has been to address uncertainty in space and time, modeling the effect of tasks being completed, instead of imposing hard specifications at fixed times, start and finish. We will present recent results that extend Schrödinger’s paradigm in such a direction. Our work is developed in the context of Markov chains and random walks on graphs. Specifically, we examine the case where one marginal distribution represents the initial state occupation of a Markov chain, while other marginals represent first-arrival time distributions at absorbing states signifying completion of tasks. We establish that when the prior is Markov, a posterior Markov policy is optimal apropos to a likelihood cost that follows Schrödinger’s dictum. We then provide a systematic method to obtain such optimal posterior law by suitably modifying prior transition probabilities. The below expository on which the presentation will be based follows Eldesoukey and Georgiou (2023).

AMS classification: 90B15, 60F10, 60J10

Keywords: Stochastic Modeling and Stochastic Systems Theory, Nonlinear Systems and Control, Nonlinear Filtering and Estimation
Abstract: This paper considers the problem of constructing finite-dimensional state space realizations for stochastic processes that can be represented as the outputs of a certain type of a causal system driven by a continuous semimartingale input process. The main assumption is that the output process is infinitely differentiable, where the notion of differentiability comes from the functional Itô calculus introduced by Dupire as a causal (nonanticipative) counterpart to Malliavin's stochastic calculus of variations. The proposed approach builds on the ideas of Hijab, who had considered the case of processes driven by a Brownian motion, and makes contact with the realization theory of deterministic systems based on formal power series and Chen-Fliess functional expansions.

Keywords: Stochastic Modeling and Stochastic Systems Theory, Nonlinear Systems and Control, Systems on Graphs
Abstract: The proof of convergence for the heterogeneous Deffuant-Weisbuch (DW) model in bounded-confidence opinion dynamics is a long-standing open problem. We resolve this problem completely, and generalize the convergence result to high-dimensional DW model with heterogeneous weighting factors.

Keywords: Stochastic Modeling and Stochastic Systems Theory, Systems Biology, Linear Systems
Abstract: We study how sequestration mechanisms, such as protein binding to decoy sites and the reversible formation of phase-separated condensates, alter the statistics of protein levels. To implement such a type of regulation, we consider simple birth-death processes in which protein molecules can reversibly switch between active (free-molecule) and inactive (sequestered) configurations with arbitrary concentration-dependent rates. When these transition rates depend only on the levels of the inactive protein, we show analytically that the steady-state distribution of the active protein level has Poisson statistics regardless of the details of the sequestration process. When transition rates depend on the active protein, the molecule copy number distribution is non-Poisson. We illustrate this deviation with the example of an enzyme-driven post-translational modification of an active protein into an inactive state, and we confirm the results using both analytical approximations and exact stochastic simulations. In summary, our results characterize the role of sequestration mechanisms in modulating the stochasticity of biochemical processes and identify noise-invariant processes.

Keywords: Coding Theory, Information Theory
Abstract: We propose a new variant of the McEliece cryptosystem. The generator matrix of the secret code is masked with polynomial matrices. The public key is then the generator matrix of a convolutional code. The plaintext is a sequence of vectors and the ciphertext is the corresponding codeword with some errors added through the sequence satisfying some weight restriction that permits to decode efficiently. We propose the use of a Generalized Reed-Solomon code as the secret code and construct the polynomial matrices to obtain a secure scheme. We analyze its security and compare the obtained results with some of the code-based public-key cryptosystems proposed for the NIST post-quantum cryptography standardization project.

Keywords: Machine Learning and Control, Optimization : Theory and Algorithms, Artificial Intelligence
Abstract: We study optimality for the safety-constrained Markov decision process which is the underlying framework for safe reinforcement learning. Specifically, we consider an undiscounted safety-constrained Markov decision process subject to random stopping times. The decision maker's goal is to reach a goal state while avoiding unsafe states with certain probabilistic guarantees. Therefore the underlying Markov chain for any control policy will be Multichain or non-ergodic since by definition there exists a goal set and an unsafe set. Bellman's principle of optimality does not hold for such a safety-constrained Markov decision process in a Multichain setting as highlighted by a counterexample. We resolve the aforementioned counterexample by considering a zero-sum game setting between the policy and the Lagrange multiplier vector. Under suitable assumptions regarding the existence of admissible policy, we propose an off-policy RL algorithm for learning an optimal policy that satisfies the probabilistic safety guarantees. After that, we present the finite time error bound of the proposed RL algorithm. Lastly, we present simulation results of the aforementioned RL algorithm on a robot in a grid world setting.

Keywords: Multidimensional Systems, Neural Networks
Abstract: From the perspective of control theory, convolutional layers (of neural networks) are 2-D (or N-D) linear time-invariant dynamical systems. The usual representation of convolutional layers by the convolution kernel corresponds to the representation of a dynamical system by its impulse response. However, many analysis tools from control theory, e.g., involving linear matrix inequalities, require a state space representation. For this reason, we explicitly provide a state space representation of the Roesser type for 2-D convolutional layers with c_mathrm{in}r_1 + c_mathrm{out}r_2 states, where c_mathrm{in}/c_mathrm{out} is the number of input/output channels of the layer and r_1/r_2 characterizes the width/length of the convolution kernel. This representation is shown to be minimal for c_mathrm{in} = c_mathrm{out}. We further construct state space representations for dilated, strided, and N-D convolutions.

Keywords: Networked Control Systems, Linear Systems, Information Theory
Abstract: In this work, we ask what the minimum data-rate is for a possibly nonlinear control law acting on a linear system to emulate the closed-loop behavior of a desired linear control law. This result gives an implicit estimate for the information flow in the feedback path of the closed-loop linear system. We formally introduce the notion of emulation and control law, and we present a data-rate theorem. Remarkably, we note that the minimum data-rate varies discontinuously with the parameters of the open and closed-loop systems. This feature is new since the usual data-rate theorem, which only requires stabilization instead of emulation, gives a continuously varying minimum data-rate.

Keywords: Optimization : Theory and Algorithms, Systems on Graphs, Networked Control Systems
Abstract: We propose a novel cooperative dynamic game with the use of distributed optimization methods. Each coalition has an objective function and the goal of all the coalitions is to optimize the summation of their objectives given the interactions of their agents. The internal interactions are described by a multigraph but we will use directly a stochastic matrix to describe them while the external interactions are described by a hypergraph. The objective functions are quadratic and they encompass information for both types of interactions to highlight their importance in the outcome for the coalitions. The problem is formulated by means of distributed optimization where the actions of the agents at each iteration are the optimum solutions of the sequential optimization problem and the game is evolving by using a tilted matrix mechanism. We show that the optimization problem at each iteration can be solved in one step via a dual decomposition algorithm by exploiting the hypergraph stucture of the optimization problem. For the complete description of the dynamic game we propose a switched dynamical system and we demonstrate its convergence numerically.

Keywords: Delay Systems, Feedback Control Systems, Networked Control Systems
Abstract: The study addresses output feedback controller synthesis for networked control systems with both stochastic multiplicative uncertainties and intrinsic channel-wise time delays. Building on the mean-square stability condition obtained, a fundamental necessary and sufficient condition for the mean-square stabilizability is derived. This criterion elucidates that an open-loop unstable time-delay system can achieve mean-square stabilization through output feedback in the presence of networked uncertainties.

Keywords: Large Scale Systems, Networked Control Systems, Linear Systems
Abstract: We study the error amplification in a vehicle platoon employing a bidirectional control strategy. By deriving closed-form expressions for the scalar transfer functions from the lead vehicle's trajectory deviation to the spacing error, considering varying numbers of vehicles in the platoon, we discuss the convergence of the transfer functions to fixed points in a complex plane and the performance of error suppression evaluated by the H-infinity-norm. Our results suggest that symmetric bidirectional control strategies offer advantages in size-independent error suppression performance over asymmetric ones.

Keywords: Networked Control Systems
Abstract: This work considers the synchronization of heterogeneous pulse-coupled oscillators (PCOs), where some oscillators might be faulty or malicious. The oscillators interact through identical pulses at discrete instants and evolve continuously with different frequencies otherwise. Despite the presence of misbehaviors, benign oscillators aim to reach synchronization. To achieve this objective, a resilient synchronization protocol is developed in this work by adapting the real-valued mean-subsequence reduced (MSR) algorithm to pulse-based interactions. We establish sufficient conditions on the initial states and graph structure under which resilient synchronization is achieved. Specifically, the normal oscillators can either detect the presence of malicious nodes or synchronize among themselves in the presence of stealthy attacks

Keywords: Networked Control Systems, Hybrid Systems, Linear Systems
Abstract: Often in practical applications, such as coordinated motion of autonomous vehicles, multi-agent systems (MASs) utilize information obtained from sensors to accomplish complex tasks, asynchronously. In this work, we consider the problem of consensus where the agents obtain relative-state measurements, at their own sampling frequencies, and employ a distributed event-triggered protocol to dictate when to update control. For the designed event-triggered protocol, only local intermittent relative-state measurements are utilized, where they are obtained and evaluated only at pre-determined event-monitoring instants; these instants are governed by sampling periods whose bounds are explicitly pre-computed, individually, for each agent. Hence, the designed protocol is inherently asynchronous and avoid Zeno behaviour by construction. To cope with the continuous-time dynamics of the agents and discrete-time sensing and controller updates, the overall MAS is modelled using the hybrid system framework. A numerical example is provided to clearly demonstrate the effectiveness of the designed protocol.

Keywords: Networked Control Systems, Stability, Transportation Systems
Abstract: This work studies the string stability of a platoon of autonomous vehicles, where the leader broadcasts information to the followers. We assume that the leader's information is susceptible to data loss and examine how this affects the string stability property of the platoon. Two lossy scenarios are considered, both with a suitable data-loss compensation strategy. In the first scenario, all vehicles in the string are subject to the same loss. In the second scenario, transmission, and we assume that the probability of data loss depends on the distance between each vehicle and the leader. We numerically observe that data loss does not seem to impact the string stability of the platoon in the first scenario under consideration. However, in the second scenario, the function governing data loss based on position within the platoon can induce string instability.

Keywords: Systems on Graphs, Networked Control Systems
Abstract: This work explores the scalability of graph networks through the so-called “plug-and-play” connections, aiming to expand a network by gradually adding nodes with specified connecting degrees. Using a resource allocation approach, we establish a tighter upper bound on the number of connectable nodes under limited connecting capacity. We also show that the bound can be reached through a ‘minimum strategy’ where added nodes prioritize establishing connections with nodes that have the smallest degrees.

Keywords: Infinite Dimensional Systems Theory

Keywords: Algebraic Systems Theory, Applications of Algebraic and Differential Geometry in Systems Theory, Nonlinear Systems and Control
Abstract: We investigate controlled invariant varieties for second order input-affine control systems with polynomial nonlinearity. For autonomous, first order systems, a variety V is said to be invariant if any trajectory starting in V remains in V for all times. We discuss how to adapt this notion in the second order case and give conditions to decide whether a variety is invariant for a second order system using symbolic computation. This results serve as a foundation to characterise controlled invariant varieties, i.e., varieties that can be rendered invariant by a polynomial state feedback.

Keywords: Algebraic Systems Theory, Linear Systems, Computations in Systems Theory
Abstract: We present the theory of linear systems over various kinds of finite commutative rings. Since these systems are finite, the trajectories have to run into repeating cycles eventually. This periodic behavior is the main interest of this topic. Using an approach similar to Fitting's lemma, a bijective-nilpotent decomposition of the system can be achieved, which in some cases even gives a decomposition of the system matrix. In particular, this allows us, to apply results about invertible system matrices, where all trajectories are purely periodic, to the more general setting. Finally, the algorithmic potential of the theory is discussed.

Keywords: Algebraic Systems Theory, Nonlinear Systems and Control, Operator Theoretic Methods in Systems Theory
Abstract: We collect three instances where the theory of generalized functions may still make contributions to the study of signals and systems. In the first, a purely algebraic approach is presented for LTI-ODE's, in terms of two operators, D and T, respectively the differentiation operator and the multiplication-by-the-independent-variable operator. This formalism adds simplicity, a duality theory, and nicely generalizes to other classes of operator equations and their solutions. In the second part we extend the classical bilateral Laplace transform to include Bohl functions with support in R by invoking Sato's hyperfunctions. Finally, in the third case we use the Colombeau algebra to allow for products of generalized functions. This is important in the study of (smooth) nonlinear systems driven by impulsive inputs, and hybrid system theory.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Nonlinear Filtering and Estimation
Abstract: Internal model principles provide necessary conditions on the structure of controllers or observers that achieve a certain control or estimation goal. In formulating such principles, care must be taken in clearly defining the properties of the plant under consideration as well as the set of candidate controllers or observers. We introduce a novel system property for non-linear state-space systems on manifolds that we call repeatability and show that this property is implied by the usual notion of controllability but is significantly weaker. In fact, we show that such a system is repeatable if and only if the reachability relation is an equivalence relation. Sontag calls this latter property reversibility. We then show that any asymptotic observer for a repeatable plant contains a full internal model of the plant, as long as the observer has been constructed on a manifold that is a surjective fiber bundle with compact fibers over the plant state manifold. We discuss in what sense the restriction to surjective fiber bundles is necessary and what happens in the case of non-compact fibers. A direct consequence of this result is that the right-hand side of such an observer can be split without loss of generality into a lift of the plant plus a correction term that is zero along lifted plant trajectories. This split might require a modification of the observer equation that does not change the error trajectories; this is the precise meaning of without loss of generality in this context. The result shows that the generalized Luenberger construction for observers is indeed most general in this setting, in the sense that all possible asymptotically stable error behaviors can be achieved by such designs.

Keywords: Linear Systems, Algebraic Systems Theory, Stability
Abstract: A constructive design of a stable left inverse for single-input linear systems is proposed. Based on structural properties relating left inversion and feasability of input reconstruction, the input observer is realized within the state-space approach, for arbitrary initial condition. While previous design is concerned with a nominal plant, that is with known parameters, we propose in a second step to apply this inversion methodology to a simplified identification problem. Reformulating the unknown parameters as a fictitious unknown input, convergent parameter estimates are retrieved from measurement. The overall methodology is illustrated by simulations.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Optimization : Theory and Algorithms, Machine Learning and Control
Abstract: We present a geometric framework for the processing of SPD-valued data that preserves subspace structures and is based on the efficient computation of extreme generalized eigenvalues. This is achieved through the use of the Thompson geometry of the semidefinite cone. We explore a particular geodesic space structure in detail and establish several properties associated with it. Finally, we review a novel inductive mean of SPD matrices based on this geometry.

Keywords: Machine Learning and Control, Artificial Intelligence, Physical Systems Theory
Abstract: We present a structure-preserving kernel method for the learning of Hamiltonian systems. In the presentation, we shall start by establishing reproducing properties of differentiable kernels on any subsets of R^{2d}, which enables us to embed the corresponding RKHS into the space of bounded differentiable functions with bounded derivatives. We then study the Hamiltonian learning problem using a kernel ridge regression, we provide an operator-theoretical framework to represent the structure-preserving kernel estimators, and we prove convergence results and error bounds for them. Finally, we present some numerical experiments.

Keywords: Machine Learning and Control, Artificial Intelligence, Neural Networks

Keywords: Nonlinear Filtering and Estimation, Stochastic Modeling and Stochastic Systems Theory, Stochastic Control and Estimation
Abstract: In this paper, we consider the collaborative attitude estimation problem of a multi-agent system. The agents are equipped with sensors that provide directional measurements and relative attitude measurements. We present a bottom-up approach where each agent runs an extended Kalman filter (EKF) locally using directional measurements and augments this with relative attitude measurements provided by neighbouring agents. The covariance estimates of the relative attitude measurements are geometrically corrected to compensate for relative attitude between the agent that makes the measurement and the agent that uses the measurement before being fused with the local estimate using the convex combination ellipsoid (CCE) method to avoid data incest. Simulations are undertaken to numerically evaluate the performance of the proposed algorithm.

Keywords: Stochastic Control and Estimation

Keywords: Model Predictive Control, Optimal Control

Keywords: Delay Systems

Keywords: Algebraic Systems Theory, Discrete Event Systems, Nonlinear Systems and Control
Abstract: We consider the growth rate of a switching max-min-plus-scaling (S-MMPS) system in a discrete-event framework. We show that an explicit, time-invariant, monotone, and arbitrarily switching MMPS system has a bounded growth rate. Further, we propose a mixed-integer linear programming problem to calculate the estimates of the smallest upper bound and the largest lower bound of the growth rate of an S-MMPS system.

Keywords: Algebraic Systems Theory, Multidimensional Systems, Linear Systems
Abstract: For a class of discrete 2D systems, where one of the independent variables partitions the plane into ‘past’ and ‘present’, called time-relevant 2D systems, we provide a necessary and sufficient condition on the trajectory data to identify the generating system, i.e., we give a necessary and sufficient condition for the data to be informative for identification. Moreover, we provide a data-driven representation of the behaviour and use it to simulate the admissible trajectories for arbitrary intervals using an informative data and initial conditions without identifying the system.

Keywords: Algebraic Systems Theory, Optimization : Theory and Algorithms, Robust and H-Infinity Control
Abstract: This paper introduces the multiple output response decoupling (MORD) theorem and applies it to control mass-chain systems. The dynamics of mass-chain systems are coupled, i.e., improving the performance of one mass might compromise those of others. Therefore, we develop the MORD theorem, which allows the modification of all system outputs independently and simultaneously without compromises. Furthermore, controlling a particular mass requires only three actuators on the consecutive masses. Finally, we apply a five-mass system to demonstrate the effectiveness of MORD control for mass-chain system control.

Keywords: Nonlinear Systems and Control, Algebraic Systems Theory, Feedback Control Systems
Abstract: The affine feedback connection of SISO nonlinear systems modeled by Chen–Fliess series is shown to be a group action on the plant which is isomorphic to semi-direct product of shuffle and additive group of non-commutative formal power series. The additive and multiplicative feedback loops in an affine feedback connection are thus proven to be structurally non-commutative. A flip in the order of these loops results in a net additive feedback loop.

Keywords: Linear Systems, Stability, Large Scale Systems
Abstract: For linear control systems, we study algebraic stability domains, which are defined by polynomial inequalities in the complex plane. Based on a characterization via generalized Lyapunov matrix inequalities, we define a suitable pair of Gramians and a balancing transformation. For special cases, we analyze the preservation of stability domains by balanced truncation.

Keywords: Networked Control Systems, Stability, Feedback Control Systems
Abstract: It is known that designing a scalable controller for multi-agent systems with third- order integrator dynamics is not possible when only relative information is available. We explore what conditions are needed if also absolute feedback is included in the control design. In this note we study how positional absolute feedback, positional relative feedback and velocity relative feedback are restricted by the relative and absolute feedback of velocity and acceleration. Contrary to intuition, this means that introducing too much relative velocity feedback, relative positional feedback or absolute positional feedback may make a controller lose its scalable stability.

Keywords: Stability, Applications of Algebraic and Differential Geometry in Systems Theory
Abstract: We investigate global dynamic feedback stabilization from a topological viewpoint. In particular, we consider the general case of dynamic feedback systems, whereby the total space (which includes the state space of the system and of the controller) is a fibre bundle, and derive conditions on the topology of the bundle that are necessary for various notions of global stabilization to hold. This point of view highlight the importance of distinguishing trivial bundles (think of a cylinder) and twisted bundles (think of a Mobius strip) in the study of global dynamic feedback stabilization, as we show that dynamic feedback defined on a twisted bundle can stabilize systems that dynamic feedback on trivial bundles cannot.

Keywords: Stability, Linear Systems
Abstract: For compact sets of complex matrices it is shown that there always exists a similarity transformation such that in the transformed set all entries of all matrices are bounded in absolute value by the joint spectral radius. The key tool for this is that every extremal norm of a matrix set has an Auerbach basis. The result implies in particular that all diagonal entries, or equivalently all one-dimensional principal submatrices, are upper bounded by the joint spectral radius. It is shown that the corresponding statement for higher dimensional principal submatrices is false. More precisely, there are finite matrix sets, such that for all points in the similarity orbit the joint spectral radii of all higher dimensional principal submatrices are strictly larger than the joint spectral radius of the original matrix set.

Keywords: Robust and H-Infinity Control, Nonlinear Systems and Control
Abstract: Redheffer's lemma is a key result in H-inf control, reducing 4-block problems to equivalent 2-block ones. This note extends its scope to potentially nonlinear controllers and offers an input/output stability analysis, avoiding the need to use realization-dependent Lyapunov stability arguments.

Keywords: Quantum Control, Coding Theory, Stability
Abstract: In this paper, we deal with the stability of the improved quantum coding method based on Markovian dissipative dynamics that is robust against thermal noise. This coding method was proposed by Nishino et al. and the suppression of the effect of thermal noise was demonstrated by numerical simulations, however the mathematical proof was not given. In this paper, we analyze the effect of the thermal noise in this method and mathematically prove that the quantum states converge to a neighborhood around the target code words. The bound of the neighborhood set is also given strictly. We also demonstrate numerical simulations, which confirm the theoretical results.

Keywords: Quantum Control, Nonlinear Filtering and Estimation, Stochastic Modeling and Stochastic Systems Theory
Abstract: We consider an interacting quantum particles system undergoing continuous imperfect observations. By transitioning to the MF limit, we study properties in particular the well-posedness of the limit equation describing the evolution of a particle subjected to a MF potential. We illustrate through an example the application of this limit equation in the design of a stabilizing feedback control.

Keywords: Quantum Control, Operator Theoretic Methods in Systems Theory, Applications of Algebraic and Differential Geometry in Systems Theory
Abstract: Finding quantum strategies for games ultimately boils down to solving systems of equations in matrix unknowns. The talk concerns a particular class of quantum games which have a long tradition for illuminating new phenomena: XOR games. A particular 2 player XOR games, the CHSH game, demonstrated the existence of quantum entanglement, subsequently Tsirelson showed how to compute an optimal quantum strategy for any 2XOR game (a semidefinite program). This talk focuses on a class of three player games called 3XOR games. There has been much recent progress on understanding perfect strategies for 3XOR games and this talk will focus on that.

The work is joint with Adam Bene Watts, Daniel Kane, Igor Klep, and Zehong Zhang.

Keywords: Infinite Dimensional Systems Theory

Keywords: Operator Theoretic Methods in Systems Theory, Mathematical Theory of Networks and Circuits, Stochastic Modeling and Stochastic Systems Theory

Keywords: Multidimensional Systems, Linear Systems
Abstract: We put forward the basic ideas for solving the following problem. Consider a two- dimensional (2D) system Σ defined over the whole grid Z2 and described by a first order (2D) model of the Fornasini-Marchesini type. Which initial conditions can be assigned on the boundary F of the first (discrete) quadrant (or quarter-plane) Q(0,0) in order to compute the system signals (inputs, local states and outputs) on Q(0,0)? Due to the nature of the updating equations of the Fornasini-Marchesini model, this would not even be a problem if Σ were defined only over the quarter-plane Q(0,0): in that case, both the inputs and the (local) states would be freely assignable on F. However, the requirement that the signal values computed on Q(0,0) correspond to Σ signals, which are necessarily defined over the whole 2D grid makes this problem nontrivial.

Keywords: Multidimensional Systems, Linear Systems, Stability
Abstract: In this paper, we further study the effective computation of stabilizing controllers of multidimensional systems. Within the algebraic analysis approach, the stabilization problem can be characterized by the fact that a certain finitely presented A-module M, naturally associated with the multidimensional system, is projective, where A denotes the ring of multivariate rational functions without poles in the closed complex unit polydisc D^n. This condition can be reduced to the existence of an element s of a polynomial ideal I which has no zero in D^n. According to the Polydisc Nullstellensatz, the latter condition is equivalent to the fact that no complex zero of the elements of I belongs to D^n. If this condition is satisfied, using cyclic resultants and linear programming, we then propose a method to compute such a polynomial s. Finally, using computer algebra methods for effectively handling basic operations on R[s^{-1}]-modules, where R is a polynomial ring, we show how to compute stabilizing controllers.

Keywords: Multidimensional Systems, Stability, Algebraic Systems Theory
Abstract: The aim is to provide data-based conditions for checking stability of strongly autonomous nD systems. It is known that because of the lack of natural ordering of the domain, the problem of defining and studying stability of nD systems is not straightforward. We focus on the following two notions of stability: (i) stability with respect to a given direction, and more generally, (ii) conic stability, that is, stability with respect to subsets of the domain having the structure of a cone. For a strongly autonomous nD system, these two notions of stability have been studied in the literature from a model based perspective. Here, we analyze their data-based counterparts. We assume that the data is noise-free, and that it is measured from a linear system of partial difference equations, with constant real coefficients, having n independent variables. We further assume that the data-generating system is a strongly autonomous nD system, that is, the solution trajectories can be uniquely determined using finite ``initial conditions''. Given a finite set of data, that is, a finite set of multi-indexed sequences, we first define a notion of informativity of nD trajectories. Using informative nD trajectories, we provide data-based tests for checking stability of nD systems with respect to a given direction, and, more generally, for conic stability.

Keywords: Machine Learning and Control
Abstract: We discuss the theories underlying positive definite and positively decomposable invariant kernels on homogeneous spaces. These provide a path for the application of reproducing kernel Hilbert space and reproducing kernel Krein space methods on non-Euclidean data spaces, respectively.

Keywords: Optimization : Theory and Algorithms
Abstract: Regularization techniques are frequently deployed in optimization-based approaches for solving inverse problems. These techniques help to promote certain desired structure in solutions. In this paper, we ask: Given a distribution, what is the optimal regularizer for data drawn from the distribution?

By restricting to regularizers that are positively homogenous, the functions we consider are in one-to-one correspondence with star bodies. We leverage the dual Brunn-Minkowski theory to show that the optimal choice of regularizer can be described as a radial function that serves as a sufficient statistic of the distribution.

Keywords: Quantum Control, Nonlinear Systems and Control, Optimization : Theory and Algorithms