We study exact controllability problems for some nonlinear systems with linear controls. Our tools are contraction fixed point theorems and nonlinear semigroup properties. We show that under the assumptions of low order nonlinearity, reversibility and the existence of certain feedback controls, the nonlinear system is exactly controllable. The constructive aspect of the theory allows the application of numerical simulation. An analog-digital realization diagram is discussed. Accurate numerical schemes are developed and error estimates are presented with concrete examples to illustrate the theory.

Chen, G., Mills, W., & Crosta, G.F. (1981). Exact Controllability Theorems and Numerical Simulations for Some Non-Linear Differential-Equations. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 19(6), 765-790 [10.1137/0319050].

Exact Controllability Theorems and Numerical Simulations for Some Non-Linear Differential-Equations

CROSTA, GIOVANNI FRANCO FILIPPO
Ultimo
1981

Abstract

We study exact controllability problems for some nonlinear systems with linear controls. Our tools are contraction fixed point theorems and nonlinear semigroup properties. We show that under the assumptions of low order nonlinearity, reversibility and the existence of certain feedback controls, the nonlinear system is exactly controllable. The constructive aspect of the theory allows the application of numerical simulation. An analog-digital realization diagram is discussed. Accurate numerical schemes are developed and error estimates are presented with concrete examples to illustrate the theory.
Si
Articolo in rivista - Articolo scientifico
Scientifica
nonlinear dynamical systems; linear control; Banach spaces; controllability via stabilisability; backward equation; global contraction mapping; non-linear wave equation; numerical algorithms; convergence rates; error estimates.
English
ZentralBlatt fuer Mathematik: review Zbl 0469.93016 signed by Ruth F. Curtain (Groningen, NL) ``The authors consider exact controllability of nonlinear differential equations (but linear in the control variable) which are defined on a Banach space. The nonlinearity is described by means of a nonlinear semigroup which also needs to be defined for t < 0. This reversibility assumption eliminates several classes of systems, for example parabolic or delay systems. The idea is to achieve exact controllability via stabilisability assumptions, neither of which can be verified in general. The control is the sum of a linear feedback of the state plus a linear feedback of a time-reversed conjugate system, which would have to be simulated. Applications are considered to some carefully chosen nonlinear ordinary differential equations and nonlinear wave equations. Finally, the authors consider a numerical algorithm to find such a control which achieves a desired terminal state, for the special case of nonlinear differential equations. They claim it works well, and include several computer aided graphs to make this point.'' --- Mathematical Reviews: review MR0634953 (84e:93011) based on authors' summary.
Chen, G., Mills, W., & Crosta, G.F. (1981). Exact Controllability Theorems and Numerical Simulations for Some Non-Linear Differential-Equations. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 19(6), 765-790 [10.1137/0319050].
Chen, G; Mills, W; Crosta, G
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10281/93757
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