Jacobi conditions and the Riccati equation for a broken extremal. (English. Russian original) Zbl 0968.49016

Osmolovskii, N. P.; Lempio, F.

J. Math. Sci., New York 100, No. 5, 2572-2592 (2000); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat Obz. 60, 187-215 (1998).

Normally, the classical computation of variations dealing with second order necessary and sufficient condition is used for the case of smooth extremal. In the paper there is proved that the classical second-order conditions formulated as a conjugate point and Riccati equation may be generalized to the case of broken extremal. The subject of this work is the following simple controlled object: \[ \dot{x}(t)=u(t), \text{ almost everywhere} \]
\[ x(0)=b_{0}, \quad x(T)=b_{1} \]
\[ (x(t), u(t),t) \in Q, \text{ almost everywhere} \] where \(x : [0, T] \rightarrow \mathbb R^n\) is an absolutely continuous function, \(x(\cdot)\in W^{1,1}\left([0,T], R^{n}\right)\) and \(u(t):[0,T]\rightarrow \mathbb R^n\) is bounded and measurable, \(u(t) \in L^{\infty}\left([0,T], \mathbb R^n\right)\). The results, although received in classical theoretical way, may be used in practical applications of optimal control theory.

Reviewer: Wladyslaw Hejmo (Kraków)

MSC:

49K15	Optimality conditions for problems involving ordinary differential equations
93C15	Control/observation systems governed by ordinary differential equations

Keywords:

calculus of variations; optimal control theory; second-order conditions; conjugate point; Riccati equation; broken extremal

References:

[1]	A. V. Dmitruk, ”Euler-Jacobi equation in the calculus of variations,”Mat. Zametki,20, No. 6, 847–858 (1976). · Zbl 0356.49009
[2]	A. V. Dmitruk, ”Jacobi-type conditions in the Bolza problem with inequalities,”Mat. Zametki,35, No. 6, 813–827 (1984). · Zbl 0552.49023
[3]	R. Henrion,La Theorie de la Variation Seconde et Ses Applications en Commande Optimal, Academie Royal de Belgique, Bruxelles-Palais des Academies (1979).
[4]	A. D. Ioffe and V. M. Tikhomirov,Theory of Extremal Problems, [in Russian], Nauka, Moscow (1974).
[5]	A. A. Milyutin and N. P. Osmolovskii,Calculus of Variations and Optimal Control, American Mathematical Society, Translations of Mathematical Monographs, Vol. 180 (1998). · Zbl 1331.49007
[6]	A. Nowakowski,A Second Order Sufficient Condition for Optimality in Nonlinear Control–The Conjugate Point Approach, Birkhäuser Verlag, Basel. · Zbl 0940.49022
[7]	N. P. Osmolovskii, ”Higher-order necessary and sufficient conditions for the Pontryagin and boundedly strong minimum in an optimal control problem,”Dokl. Akad. Nauk SSSR,303, 1052–1056 (1988).
[8]	N. P. Osmolovskii, ”Quadratic conditions for nonsingular extremals in optimal control (a theoretical treatment),”Russian J. Math. Phys.,2, No. 4, 487–516 (1994). · Zbl 0916.49016
[9]	N. P. Osmolovskii, ”Quadratic conditions for nonsingular extremals in optimal control (examples),”Russian J. Math. Phys.,5, No. 3, 373–388 (1998).
[10]	E. E. Shnol’, ”On degeneration in the simplest problem of the calculus of variations,”Mat. Zametki,24, No. 5 (1978).
[11]	M. G. Tagiev, ”A necessary and sufficient condition for the strong extremum in a degenerate problem of the calculus of variations,”Usp. Mat. Nauk,34, No. 4, 211–212 (1979). · Zbl 0474.49022