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The Morse index and the Maslov index for extremals of controlled systems. (English. Russian original) Zbl 0614.58016

Sov. Math., Dokl. 33, 392-395 (1986); translation from Dokl. Akad. Nauk SSSR 287, 521-524 (1986).
Let us consider a controlled nonstationary system of the form (*) \(\dot m=f_ t(m,u)\) where \(m\in M\), a \(C^{\infty}\)-differentiable manifold. Let \(p_ t(m)\) be the flow corresponding to \(u=0\) and \(F_ t: u\mapsto m(t)\) the map which associates with each admissible control the solution of (*) corresponding to a fixed initial state \(m_ 0\). Critical points of the map \(G_ t=p_ t^{-1}\circ F_ t\) are called extremals. The main purpose of this paper is to give an explicit expression for the Morse index of the Hessian of \(G_ t\) at an extremal. The importance of these indices is well-known in control theory and in the calculus of variations.
Reviewer: A.Bacciotti

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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