Asymptotic controllability and optimal control

Journal of Functional Analysis, 1990

ESAIM: Control, Optimisation and Calculus of Variations, 2013

The paper deals with deterministic optimal control problem with state constraints and non-linear dynamics. It is known for such a problem that the value function is in general discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

Funkcialaj Ekvacioj, 1994

Applied Mathematics & Optimization, 1991

We present two convergence theorems for Hamilton Jacobi equations and we apply them to the convergence of approximations and perturbations of optimal control problems and of two-players zero-sum differential games. One of our results is, for instance, the following. Let T and T h be the minimal time functions to reach the origin of two control systems y' = f(y, a) and y' = fh(Y, a), both locally controllable in the origin, and let ~ be any compact set of points controllable to the origin. If 11 fhf II ~ < Ch, then IT(x)-Th(x)l < Cgh", for all x ~ o,~, where ~ is the exponent of H61der continuity of T(x).

IFAC Proceedings Volumes, 1992

SIAM Journal on Control and Optimization, 1989

Optimal control problems, with no discount, are studied for systems governed by nonlinear "parabolic" state equations, using a dynamic programming approach. If the dynamics are stabilizable with respect to cost, then the fact that the value function is a generalized viscosity solution of the associated Hamilton-Jacobi equation is proved. This yields the feedback formula. Moreover, uniqueness is obtained under suitable stability assumptions. Key words, optimal control, Hamilton-Jacobi equations, viscosity solutions, evolution equations, unbounded operators AMS(MOS) subject classifications. 49C20, 34G20 *

2008

This lecture will highlight the contributions of optimal control theory to geometry and mechanics. The basic object of study is the reachable sets of families of vector fields parametrized by control functions. We will show how the extremal properties of the reachable sets lead to the Hamiltonians and how these Hamiltonians alter our understanding of the classical calculus of variations in which the Euler-Lagrange equation is a focal point of the subject.

Nonlinear Differential Equations and Applications NoDEA, 2020

We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set $$\mathbf{C}$$ C with$$({p_{0}},W)$$ ( p 0 , W ) -regulated cost, for some continuous, state-dependent function W and some constant $${p_{0}}&gt;0$$ p 0 &gt; 0 : it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by $$W(z)/{p_{0}}$$ W ( z ) / p 0 . Then, we show how the existence of a special, semiconcave Control Lyapunov Function W, called $${p_{0}}$$ p 0 -Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if the...

2006

In this article, we present and discuss the infinite horizon optimal control problem subject to stability constraints. First, we consider optimality conditions of the Hamilton-Jacobi-Bellman type, and present a method to define a feedback control strategy. Then, we address necessary conditions of optimality in the form of a maximum principle. These are derived from an auxiliary optimal control problem with mixed constraints.

Applied Mathematics and Optimization, 1999

In this paper we extend to completely general nonlinear systems the result stating that the H ∞ suboptimal control problem is solved if and only if the corresponding Hamilton-Jacobi-Isaacs (HJI) equation has a nonnegative (super)solution. This is well known for linear systems, using the Riccati equation instead of the HJI equation. We do this using the theory of differential games and viscosity solutions.