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Feedback stabilizability of nonlinear stochastic systems with state- dependent noise. (English) Zbl 0618.93068
The authors consider the problem of local and global stabilizability of a nonlinear system governed by the following stochastic differential equation \[ dx(t)=Ax(t)+Bu(x(t))+H(x,u(x))+\sum_{i}\sigma_ ix(t)dw_ i(t). \] It is assmed that for every real symmetric matrix \(M\geq 0\) \[ \sum_{i}\sigma_ i^ TM\sigma_ i\leq \lambda M,\quad 0\leq \lambda <\infty \] and \[ (i)\quad (A,B)\quad is\quad controllable, \]
\[ (ii)\quad H(0,u)=0\quad for\quad u\in R^ r,\quad \lim_{| x| \to 0}| H(x,u(x))| /| x| =0. \] Based on the solution of the stochastic algebraic Riccati equation, sufficient conditions are derived which assure the existence of state feedback control laws such that the closed-loop system is locally and globally stable in probability. The paper is not precisely written and it does not contain illustrative examples.

MSC:
93E15 Stochastic stability in control theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93D15 Stabilization of systems by feedback
15A24 Matrix equations and identities
93C10 Nonlinear systems in control theory
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References:
[1] AHMED N. U., Optimal Control of Stochastic Systems in Probabilistic Analysis and Related Topics 2 (1979) · Zbl 0445.60050
[2] HAS’MINSKII R, Z., Stochastic Stability of Differential Equations (1980)
[3] DOI: 10.1080/00207728608926880 · Zbl 0603.93065
[4] RUSSELL D. L., Mathematics of Finite-Dimensional Control Systems Theory and Design (1979) · Zbl 0408.93002
[5] DOI: 10.1016/0005-1098(76)90029-7 · Zbl 0329.93036
[6] WONHAM W. M., SIAM J. Control 1968 pp 681–
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