Journal of Optimization Theory and Applications

, Volume 60, Issue 1, pp 81–91

# Minimal time function and viscosity solutions

• V. Staicu
Contributed Papers

## Abstract

Two theorems in Ref. 1 are generalized. It is proved that, ifV(A,Γ) is the set of points that can be steered to the origin along a solution of the control systemx′=Ax−c, ifc(t)∈Γ, Γ is a compact subset ofR n , 0∈ intrelco Γ, and if a rank condition holds, then the minimal time functionT(·) is a viscosity solution of the Bellman equation
$$\max \{ \left\langle {DT(x),\gamma - Ax} \right\rangle :\gamma \varepsilon co\Gamma \} - 1 = 0,x\varepsilon V(A,\Gamma )\backslash \{ 0\} ,$$
and of the Hàjek equation
$$1 - \max \{ \left\langle {DT(x),\exp [ - AT(x)]} \right\rangle :\gamma \varepsilon co\Gamma \} = 0,x\varepsilon V(A,\Gamma ).$$

### Key Words

Minimal time function Bellman equation Hàjek equation viscosity solutions linear control system

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## Authors and Affiliations

• V. Staicu
• 1
1. 1.International School for Advanced Studies (SISSA)TriesteItaly