Hamilton-Jacobi-Bellman equations with fast gradient-dependence

Abstract

We investigate existence, uniqueness, and regularity properties for a class of H-J-B equations arising in non-linear control problems with unbounded controls. These equations involve Hamiltonians which are superlinear in the adjoint variable, and they have been already studied in the case when the growth in the adjoint variable is, in a sense, uniform with respect to the state variable. For instance, this is the case of the linear-quadratic problem. On the contrary, our results concern Hamiltonians that are superlinear in the adjoint variable, possibly not uniformly with respect to the state variable. Actually, this is the general situation one has to deal with when considering optimal control problems with a nonlinear dynamics (e.g. by slightly perturbing the linear quadratic problem). We also investigate situations where the fast growth of the Hamiltonian in the adjoint variable degenerates into a very discontinuity. Such Hamiltonians arise quite naturally in those optimal control problems where, roughly speaking, the dynamics and the cost display the same growth in the control variable. on the data f , , and g : (A 1) f : [0,T ] × R n × R m → R n is continuous and, for every compact subset Q ⊂ [0,T ] × R n there exists a positive constant L and a modulus ω f verifying |f (t 1 ,x 1 ,c) − f (t 2 ,x 2 ,c)| ≤ (1 + |c| α)(L|x 1 − x 2 | + ω f (|t 1 − t 2 |), for all (t 1 ,x 1 , c), (t 2 ,x 2 ,c) ∈ Q × R m (by modulus we mean a positive, nondecreasing function, null and continuous at zero).

State-Constrained Stochastic Optimal Control Problems via Reachability Approach

“…Therefore, to give a precise meaning to the HJB equation satisfied by w, we will use an approach based on some ideas presented in [13] and exploited also in [6]. We point out that similar ideas have been introduced in [32,33] for deterministic control systems and in [29] for stochastic problems where the unbounded control acts only on the drift.…”

A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable

“…Regularity of solutions of Hamilton-Jacobi equations with superlinear growth have been the object of several works (see in particular Lions [6], Barles [2], Rampazzo and Sartori [7]). Our aim is to show that u is locally Hölder continuous with Hölder exponent and constant depending only M , δ, q and T .…”

Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations

“…We recall that uniqueness and existence results of unbounded solutions to a class of first order Hamilton Jacobi equations (corresponding to unbounded control problems which include the case of quadratic Hamiltonians), have been addressed by several authors, see, e.g. the book of Bensoussan [9], the papers of Alvarez [1], Bardi and Da Lio [4], Cannarsa and Da Prato [13], Rampazzo and Sartori [28] in the case of convex operators, and the papers of Da Lio and McEneaney [19] and Ishii [23] for more general operators. However most of these last results have been obtained under assumptions implying the coercivity of the Hamiltonian with respect to the gradient uniformly in the state variable, namely H(x, p) → +∞ as |p| → +∞ for all x ∈ IR N .…”