Guaranteed optimal reachability control of reaction-diffusion equations using one-sided Lipschitz constants and model reduction

We show that, for any spatially discretized system of reaction-diffusion, the approximate solution given by the explicit Euler time-discretization scheme converges to the exact time-continuous solution, provided that diffusion coefficient be sufficiently large. By "sufficiently large", we mean that the diffusion coefficient value makes the one-sided Lipschitz constant of the reaction-diffusion system negative. We apply this result to solve a finite horizon control problem for a 1D reaction-diffusion example. We also explain how to perform model reduction in order to improve the efficiency of the method

Fast numerical methods for robust nonlinear optimal control under uncertainty

This thesis treats different aspects of nonlinear optimal control problems under uncertainty in which the uncertain parameters are modeled probabilistically. We apply the polynomial chaos expansion, a well known method for uncertainty quantification, to obtain deterministic surrogate optimal control problems. Their size and complexity pose a computational challenge for traditional optimal control methods. For nonlinear optimal control, this difficulty is increased because a high polynomial expansion order is necessary to derive meaningful statements about the nonlinear and asymmetric uncertainty propagation. To this end, we develop an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision. The algorithmic contribution is complemented by a convergence proof showing that the solutions of the optimal control problem after application of the polynomial chaos method approach the correct solution for increasing expansion orders. To obtain a further speed-up in solution time, we develop a structure-exploiting algorithm for the fast derivative generation. The algorithm makes use of the special structure induced by the spectral projection to reuse model derivatives and exploit sparsity information leading to a fast automatic sensitivity generation. This greatly reduces the computational effort of Newton-type methods for the solution of the resulting high-dimensional surrogate problem. Another challenging topic of this thesis are optimal control problems with chance constraints, which form a probabilistic robustification of the solution that is neither too conservative nor underestimates the risk. We develop an efficient method based on the polynomial chaos expansion to compute nonlinear propagations of the reachable sets of all uncertain states and show how it can be used to approximate individual and joint chance constraints. The strength of the obtained estimator in guaranteeing a satisfaction level is supported by providing an a-priori error estimate with exponential convergence in case of sufficiently smooth solutions. All methods developed in this thesis are readily implemented in state-of-the-art direct methods to optimal control. Their performance and suitability for optimal control problems is evaluated in a numerical case study on two nonlinear real-world problems using Monte Carlo simulations to illustrate the effects of the propagated uncertainty on the optimal control solution. As an industrial application, we solve a challenging optimal control problem modeling an adsorption refrigeration system under uncertainty

Benelux meeting on systems and control, 23rd, March 17-19, 2004, Helvoirt, The Netherlands

Book of abstract

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Dynamical Systems; Proceedings of an IIASA Workshop, Sopron, Hungary, September 9-13, 1985

The investigation of special topics in systems dynamics -- uncertain dynamic processes, viability theory, nonlinear dynamics in models for biomathematics, inverse problems in control systems theory -- has become a major issue at the System and Decision Sciences Research Program of IIASA. The above topics actually reflect two different perspectives in the investigation of dynamic processes. The first, motivated by control theory, is concerned with the properties of dynamic systems that are stable under variations in the systems' parameters. This allows us to specify classes of dynamic systems for which it is possible to construct and control a whole "tube" of trajectories assigned to a system with uncertain parameters and to resolve some inverse problems of control theory within numerically stable solution schemes. The second perspective is to investigate generic properties of dynamic systems that are due to nonlinearity (as bifurcations theory, chaotic behavior, stability properties, and related problems in the qualitative theory of differential systems). Special stress is given to the applications of nonlinear dynamic systems theory to biomathematics and ecology. The proceedings of a workshop on the "Mathematics of Dynamic Processes", dealing with these topics is presented in this volume