Journal ArticleDOI
Optimization Techniques for State-Constrained Control and Obstacle Problems
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TLDR
In this article, a dynamic programming approach is used to design control laws for systems subject to complex state constraints, where the problem of reachability under state constraints is formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given by Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities.Abstract:
The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms of Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities. The solution of these relations is complicated in general; however, for linear systems, the value functions may be described also in terms of duality relations of convex analysis and minmax theory. Consequently, solution techniques specific to systems with a linear structure may be designed independently of HJB theory. These techniques are illustrated through two examples.read more
Citations
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Journal ArticleDOI
Hamilton–Jacobi Formulation for Reach–Avoid Differential Games
Kostas Margellos,John Lygeros +1 more
TL;DR: The main advantage of the approach proposed is that it can be applied to a general class of target-hitting continuous dynamic games with nonlinear dynamics, and has very good properties in terms of its numerical solution, since the value function and the Hamiltonian of the system are both continuous.
Journal ArticleDOI
Reachability and Minimal Times for State Constrained Nonlinear Problems without Any Controllability Assumption
TL;DR: A new continuous level-set approach for characterizing the optimal times and the backward-reachability sets is given, which leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption.
Posted Content
Hamilton-Jacobi formulation for reach-avoid differential games
Kostas Margellos,John Lygeros +1 more
TL;DR: In this article, a new framework for formulating reachability problems with competing inputs, nonlinear dynamics and state constraints as optimal control problems is developed, which can be applied to a general class of target hitting continuous dynamic games with non-linear dynamics, and has very good properties in terms of its numerical solution.
Journal ArticleDOI
Discontinuous Feedback and Nonlinear Systems
TL;DR: In this article, the controllability and stability of control systems that are nonlinear, and for which, for whatever reason, linearization fails, are investigated. And the need for nonsmooth control-Lyapunov functions is discussed.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Book
Linear Matrix Inequalities in System and Control Theory
TL;DR: In this paper, the authors present a brief history of LMIs in control theory and discuss some of the standard problems involved in LMIs, such as linear matrix inequalities, linear differential inequalities, and matrix problems with analytic solutions.