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Invariance of the approximately reachable set under nonlinear perturbations. (English) Zbl 0729.49022

The paper studies the reachability of nonlinear control systems of the form \(\Sigma\) : \(\dot x=Ax+F(x)+B(u)\) where A is some linear operator, F and B are nonlinear operators and u is a control function chosen in some set U of admissible input functions. The exact problem that is studied, is to see when some state \(\xi\) is approximately reachable for the system \(\dot x=Ax+B(u)\) (i.e. when eliminating the nonlinearity F) then also this state \(\xi\) is approximately reachable for the system \(\Sigma\). The arguments used in the paper make the work fitting in the setting of “the fixed point approach to controllability”. Some specific examples illustrate the approach. These examples include a nonlinear heat equation, a distributed control system and a second order nonlinear system.

MSC:

49K40 Sensitivity, stability, well-posedness
35B37 PDE in connection with control problems (MSC2000)
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
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