A Viscosity Approach to Infinite-Dimensional Hamilton--Jacobi Equations Arising in Optimal Control with State Constraints

Abstract

We consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions, where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous. We prove that the value function is a viscosity solution of the Bellman equation and is in fact the minimal nonnegative supersolution.

Lyapunov Functions for Infinite-Dimensional Systems

“…It was then refined and simplified by Crandall-Lions [4] and Tataru [20], see also [8,11]. Here we follow the approach of [4,8]. In order to recall this definition, first we introduce the ''test functions'' that we will use to define viscosity solutions.…”

“…We will need the following estimates on the trajectories of (1.1): Estimates (2.5) and (2.6) follow easily from the Gronwall inequality, see e.g. [8].…”

“…ð0; pÞ as rðtÞ ¼ p 2 þ tan À1 ðtÞ and note that r is strictly increasing and 1-Lipschitz continuous on R. Define also W : DðAÞ Â R ! ð0; pÞ according to W ðx; rÞ ¼ rðU ðxÞ þ rÞ: Choosing t ¼ 0 in (3.1) shows that to [8,9,13] for an extensive discussion. Note, however, that it is actually similar to Definition 2.7, but taking into account that the unbounded operator A only depends on a subset of the variables (equation with separated variables).…”

“…In finite dimensions the situation is fairly well understood, see one of the authors [18] for the latest results in this direction, and the references therein. For other results on infinite-dimensional systems such as (1.1) the reader can consult our papers [8][9][10], where this kind of approach is pursued and applied in the more general context of control theory and differential games. Note that if V is lower semicontinuous and g50 then (1.9) yields that lim sup l#0 V ðJ l xÞ4V ðxÞ for every x 2 domðV Þ, and then (1.11) automatically holds by lower semicontinuity of V .…”

Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost

“…Following [10,11], viscosity solutions of the Hamilton-Jacobi equation (2.14) with discontinuous and extended real-valued coefficients are defined as follows. Note that solutions are defined in two different ways.…”