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Time-optimal control of two-dimensional systems and regular synthesis. (English) Zbl 0753.49008

Summary: A time-optimal control of a system \(u'=v-F(u)\), \(v'=-g(u)+w(t)\), \(| w|\leq K\), \(F\), \(g\in C^ 1(R)\) is studied. Pontryagin’s maximum principle is used to prove that optimal controls are piecewise constant. An optimal feedback control is studied and a construction of a locus of switching is described. Then a regular synthesis in Boltyanskij’s sense is defined and its existence is proved in special cases. In the last part the obtained results are compared with the classical ones of P. Boltyanskij, and E. B. Lee and L. Markus.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
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References:

[1] BARBANTI L.: Lienard equations and control. Functional differential equations and bifurcations (Proceedings. Sao Carlos. Brasil 1979). Springer-Verlag, 1-22.
[2] BOLTYANSKII V. G.: Mathematical Methods of Optimal Control (Russian). Nauka, Moscow 1969
[3] BOLTYANSKII V. G.: Sufficient conditions of optimality and the justification of the method of dynamic programming. (Russian). Izv. Akad. Nauk SSSR. Seria Mat., 28, 1964, 481-514.
[4] BOLTYANSKII V. G.: Sufficient conditions of optimality (Russian). DAN SSSR 140, No. 5, 1961, 994-997.
[5] BRUNOVSKÝ P.: Every normal linear system has a regular time-optimal synthesis. Math. Slovaca 28. 1978. 81 - 100. · Zbl 0369.49013
[6] BRUNOVSKÝ P.: Existence of regular synthesis for general control problems. J. Diff. Eq. 38, 1980, 317-343. · Zbl 0417.49030
[7] BRUNOVSKÝ P.: Regular synthesis for the linear-quadratic optimal control problem with linear control constraints. J. Diff. Eq. 38. 1980, 344-360. · Zbl 0417.49014
[8] CONTI R.: Equazione di Van der Pol e controllo in tempo minimo. Rapporti del 1st. Mat. ”U. Dini”, 13 1976 77.
[9] DAVIS M. J.: A property of the switching curve for certain systems. Int. J. Control 12, 1970, 457-463. · Zbl 0195.44302
[10] JAMES E. M.: Time optimal control and the Van der Pol oscillator. J. Inst. Math. Applies. 13, 1974, 67-81. · Zbl 0275.49039
[11] KUBEN J.: Time-optimal control of two-dimensional systems. CSc. - thesis, UJEP Brno 1985. · Zbl 0753.49008
[12] KUBEN J.: Global controllability of two-dimensional systems and time-optimal control (Czech). Sborník VAAZ Brno, řada B. 2, 1986.
[13] LEE E. B., MARKUS L.: Foundations of Optimal Control Theory. J. Wiley and Sons 1967. In Russian Nauka. Moscow 1972. · Zbl 0159.13201
[14] LEE E. B., MARKUS L.: Optimal Control for Nonlinear Processes. Archive for Rational Mech. and Anal. 8. 1961. 36-58. · Zbl 0099.08703
[15] LEE E. B., MARKUS L.: On the existence of optimal controls. Trans. ASME, series D, J. of Basic Engin. 84. 1962. No 1. 13-23.
[16] LEE E. B., MARKUS L.: Synthesis of optimal control for nonlinear processes with one degree of freedom. Proceedings of Inter. Sympos. on Nonlin. Vibrations, t. III, Izdat. AN USSR. Kijev 1963, 200-218.
[17] LEE E. B., MARKUS L.: On necessary and sufficient conditions of time-optimality for nonlinear second order systems (Russian). Proceedings of 2nd Congress IFAC Basel 1963, Nauka. Moscow 1965, 155-167.
[18] VILLARI G.: Ciclo limite di Linèard e controllabilitá. Bol. Univ. Math. Hal. (5) 17-A, 1980, 406-413. · Zbl 0433.34023
[19] REISSIG R., SANSONE G., CONTI R.: Qualitative Theorie nichtlinearer Differential-gleichungen. Edizioni Cremonese. Roma 1963. In Russian Nauka. Moscow 1974. · Zbl 0114.04302
[20] NEUMAN F.: Sur les équations différentielles linéaires oscillatoires du deuxiéme ordre avec la dispersion fondamentale \(o(l) = l - \pi\). Buletinul Institutuliu Politehnic Din Iasi, X(XIV), Fasc. 1 2. 1964, 37-42. · Zbl 0151.12202
[21] NEUMAN F.: Criterion of periodicity of solutions of a certain differential equation with a periodic coefficient. Annali di Mat. pura et app.. (IV). vol. I.XXV. 1967, 385-396. · Zbl 0148.07104
[22] KUBEN J.: Establishing of Locus of Switching of Optimal Feedback Control (Czech). Sborník VAAZ Brno, rada B. 4. 1988.
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