A class of Hamilton-Jacobi equations with unbounded coefficients in Hilbert spaces. (English) Zbl 1002.35056

Shimano, K.

Appl. Math. Optimization 45, No. 1, 75-98 (2002).

This paper is concerned with the Hamilton-Jacobi equations \[ \lambda u+\langle \partial\varphi(x)+B(x),Du\rangle + F(x,Du)=0\;\text{in} S\tag{1} \] and \[ u_t+\langle \partial\varphi(x)+B(x,t),Du\rangle+F(x,t,Du)=0\;\text{in} S\times (0,T),\tag{2} \] where \(S\) is a closed subset of the real Hilbert space \(H\) which satisfies \(S\subseteq\overline{D(\partial \varphi)}\), and, moreover, is dense in \(S\). The function \(\varphi:H\to[0,\infty]\) is proper lower semicontinuous and convex, \(F\) is a real-valued function, and \(B\) is a possibly multivalued nonlinear operator. Here are studied (1) and (2) for the case where \(B\) is an unbounded (multivalued) operator. Comparison results for (1) and (2) are stated and proved in Section 2. In Section 3, by using Perrons method, the existence results for (1) and (2) are proved. For the infinite and finite horizon cases, in Section 4 is examined the fact that the value function of an optimal control problem satisfies the corresponding Bellman equation of the form (1) or (2). Finally, some examples of nonlinear parabolic equations that can be applied to optimal control problems are presented. In particular, the example for Navier-Stokes equations in three-dimensional bounded domains is introduced.

Reviewer: V.Iftode (Bucureşti)

MSC:

35K55	Nonlinear parabolic equations
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49L20	Dynamic programming in optimal control and differential games
35R15	PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
47N20	Applications of operator theory to differential and integral equations

Keywords:

Perrons method; Bellman equation