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A `universal’ construction of Artstein’s theorem on nonlinear stabilization. (English) Zbl 0684.93063
Summary: This note presents an explicit proof of the theorem - due to {\it Z. {\it Artstein}} [Nonlinear Anal., Theory Methods Appl. 7, 1163-1173 (1983; Zbl 0525.93053)] - which states that the existence of a smooth control- Lyapunov function implies smooth stabilizability. Moreover, the result is extended to the real-analytic and rational cases as well. The proof uses a `universal’ formula given by an algebraic function of Lie derivatives; this formula originates in the solution of a simple Riccati equation.

MSC:
93D15Stabilization of systems by feedback
93D20Asymptotic stability of control systems
93C10Nonlinear control systems
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References:
[1] Artstein, Z.: Stabilization with relaxed controls. Nonlinear anal., 1163-1173 (1983) · Zbl 0525.93053
[2] Dayawansa, W. P.; Martin, C. F.: Asymptotic stabilization of two dimensional real analytic systems. Systems control lett. 12, 205-211 (1989) · Zbl 0673.93064
[3] Delchamps, D. F.: Analytic stabilization and the algebraic Riccati equation. Proc. IEEE conf. Dec. and control, 1396-1401 (1983)
[4] Jurdjevic, V.; Quinn, J. P.: Controllability and stability. J. differential equations 28, 381-389 (1978) · Zbl 0417.93012
[5] Kawski, M.: Stabilization of nonlinear systems in the plane. Systems control lett. 12, 169-175 (1989) · Zbl 0666.93103
[6] Hörmander, L.: The analysis of linear partial differential operators II. (1983) · Zbl 0521.35002
[7] Sontag, E. D.: A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control optim. 21, 462-471 (1983) · Zbl 0513.93047
[8] Sontag, E. D.: Continuous stabilizers and high-gain feedback. IMA J. Math. control and inform. 3, 237-253 (1986) · Zbl 0628.93009
[9] Sontag, E. D.: Smooth stabilization implies coprime factorization. IEEE trans. Automat. control 34, 435-443 (1989) · Zbl 0682.93045
[10] Sontag, E. D.; Sussmann, H. J.: Remarks on continuous feedback. Proc. IEEE conf. Dec. and control (1989) · Zbl 0675.93064
[11] Tsinias, J.: Sufficient lyapunovlike conditions for stabilization. Math. control signals systems 2, 343-347 (1989) · Zbl 0688.93048