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Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I: Equations of unbounded and degenerate control problems without uniqueness. (English) Zbl 0955.49016
Given a differentiable function \(u: \mathbb R^n\to \mathbb R\) satisfying the differential inequality \[ Du(x)f(x) + h(x) \geq 0 \] for Lipschitz continuous functions \(f: \mathbb R^n\to \mathbb R^n\) and \(h: \mathbb R^n\to \mathbb R\), straightforward integration implies the equality \[ u(x) = \inf_{t\geq 0} \left\{u(x(t,x_0)) + \int_0^t h(x(\tau,x_0)) \,d\tau\right\}, \] where \(x(t,x_0)\) is the solution of the initial value problem \(\dot{x} = f(x)\), \(x(0,x_0)=x_0\).
This nice article shows that similar implications are true also for non-differentiable functions \(u\) which are viscosity sub- or supersolutions of the Hamilton-Jacobi-Bellman equation \[ \sup_{a\in A} \left\{ -Du(x) f(x,a)-h(x,a)+k(x,a)u(x)\right\} = 0, \] even for degenerate equations that do not admit a unique viscosity solution.

49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35F20 Nonlinear first-order PDEs
35D99 Generalized solutions to partial differential equations
49L20 Dynamic programming in optimal control and differential games