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Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V: Unbounded linear terms and \(B\)-continuous solutions. (English) Zbl 0739.49017

[For part IV see the authors, ibid. 90, No. 2, 237-283 (1990; Zbl 0739.49016).]
This paper is the fifth in a series on viscosity solutions of first order Hamilton-Jacobi equations in infinite dimensions. This particular paper treats Hamilton-Jacobi equations in a real separable infinite dimensional Hilbert space involving linear terms \(\langle Ax,\nabla u\rangle\) with \(A\) being an unbounded linear maximal monotone operator in the Hilbert space and \(\nabla u\) being the Fréchet differential of \(u\). The existence results introduce the idea of \(B\)-continuity (continuity with respect to a bounded linear oprator). Solutions are constructed via representation formulas from differential games and optimal control, and not from Perron’s method.

MSC:

49L99 Hamilton-Jacobi theories
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
49J50 Fréchet and Gateaux differentiability in optimization
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
91A23 Differential games (aspects of game theory)

Citations:

Zbl 0739.49016
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References:

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