## 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.

### MSC:

 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