On Bellman's approach to optimal control theory
According to the author, the aim of this note is to obtain a refinement of his own result in [Math. Notes 38, 658-660 (1985); translation from Mat. Zametki 38, No.2, 265-269 (1985; Zbl 0593.49025)] concerning a certain type of generalization of the Bellman's equation satisfied by the \textit{minimal-time function}: \[ T(y):=\displaystyle \inf_{x(.)}t_1(y;x(.)), \;\;y\in X, \;t_1= t_1(y;x(.))>0 \] subject to: \[ x'(t)\in P(x(t)) \text{ for a.e. } t\in [0,t_1], \;x(0)=y, \;x(t)\neq x_1 \;\forall \;t\in [0,t_1), \;x(t_1)=x_1 \] where \(x_1\in X\) is a given ``target point''. The author uses his own ``differential set of a function'' \(\phi (.):W\subset X \to R\) in a direction \(u\in X\), defined by: \[ DS_u\phi (x):=\bigcap_{\varepsilon >0}\text{Cl}[\bigcup_{\delta \in (0, \varepsilon)} \delta^{-1}[\phi ([x+\delta (u+\varepsilon B)]\cap W)-\phi (x)] \] which actually coincides with the better known ``set-valued contingent derivative'' in [\textit{J. P. Aubin} and \textit{H. Frankowska}, ``Set-valued analysis'' (1990; Zbl 0713.49021)]. The main result, expressed in terms of certain very abstract ``sets of admissible directions'' \(U(x)\subset P(x)\), \(x\in W:= \text{dom}(T(.))\), states that the minimal-time function satisfies the differential inequality: \[ \sup_{u\in U(x)}\sup DS_u(-T) (x)\leq 1 \;\forall \;x\in W:= \text{dom}(T(.)) \] which, at differentiability points of \(T(.)\), coincides with the usual Bellman's equation. Related results, proved in the slightly different context of ``extreme contingent directional derivatives'' and of some more explicitely defined ``sets of generalized tangent directions to admissible trajectories'', may be found in [\textit{Şt. Mirică}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Mat. 38, No. 1, 89-102 (1992; Zbl 0823.49020)]; one may note also that the proofs may be simplified using the well-known fact that for any admissible trajectory, the real function \(t\mapsto T(x(t))+t\) is increasing on the interval \([0,t_1]\) and it is constant along any optimal trajectory.
scientific article; zbMATH DE number 1517559