## An approach to non-linear control problems using fixed-point methods, degree theory and pseudo-inverses.(English)Zbl 0563.93013

This paper is devoted to the problem of state estimation and controllability of the system (1) $$\dot z=Az+N(z)+Bu,$$, $$z(0)=z_ 0$$, $$t\in [t_ 0,T]$$, (2) $$y=Cz$$, where $$z\in Z$$ (Z a Banach space), $$u\in U$$; A, B, C are bounded linear operators and N(z) is a nonlinear, continuous operator. The solution of (1) is understood in a mild sense. For $$B=0$$, if the linear part of the system (1)-(2) is continuously initially observable, the state of the system (1) can be constructed, given an observation y, from a ball in the space Y, on [0,T]. The proof is based on the Leray-Schauder theorem. If the linear part of the system (1) is exactly controllable on [0,T] to some subspace V (from the origin), then the state of the system (1) can be steered from the origin to any final state z(T) lying in a ball in V. The proof relies on Nussbaum’s fixed point theorem. The radius of both balls is estimated. As examples, two wave equations with nonlinear perturbations are considered.
Reviewer: Z.Wyderka

### MSC:

 93B05 Controllability 93B07 Observability 93C10 Nonlinear systems in control theory 47H10 Fixed-point theorems 55M25 Degree, winding number 55M20 Fixed points and coincidences in algebraic topology
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### References:

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