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Synthesis of time-variant optimal control with nonquadratic criteria. (English) Zbl 0872.49015

Summary: Under the assumption of convexity of nonquadratic time-variant criteria for a linear time-variant control system, it is proved in this paper that the closed-loop synthesis of the optimal control is given by a nonlinear feedback \[ u(t)= -R^{-1}(t) B^*(t)P(t,x(t)), \] in which \(P(t,x)\) is the normal solution of a quasi-Riccati operator equation. It is also shown that the nonlinear feedback operator \(P(t,x)\) can be explicitly expressed by solutions of the associated nonlinear algebraic equation and nonlinear integral equations, respectively, in three cases corresponding to Mayer problems, Lagrange problems, and Bolza problems.

MSC:

49N35 Optimal feedback synthesis
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References:

[1] Berkovitz, L. D., Optimal Control Theory (1971), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0295.49001
[2] Berger, M. S., Nonlinearity and Functional Analysis (1977), Academic Press: Academic Press New York
[3] Ekeland, I.; Temam, R., Convex Analysis and Variational Problems (1976), North-Holland: North-Holland Amsterdam
[4] Lee, E. B.; Markus, L., Foundations of Optimal Control Theory (1967), Wiley: Wiley New York · Zbl 0159.13201
[5] You, Y., Nonquadratic optimal regulators and solutions of quasi-Riccati equations, Scientia Sinica Ser. A, 30, 249-261 (1987)
[6] You, Y., A nonquadratic Bolza problem and a quasi-Riccati equation for distributed parameter systems, SIAM J. Control Optim., 25, 905-920 (1987) · Zbl 0632.49004
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