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Extremals of the time optimal control problem for a material point moving along a straight line in the presence of friction and limitation on the velocity

Osmolovskii N. P., Figura A., Kośka M., Wójtowicz M.

Control and Cybernetics

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2017

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Vol. 46, no. 4

305--324

EN

This paper provides an analysis of the time optimal control problem for a material point moving along a straight line in the presence of strength of resistance to movement (friction) and subject to constraint on the velocity. The point is controlled by a limited traction or braking force. The analysis of the problem is based on the maximum principle for state constraints in the Dubovitskii-Milyutin form, see Dubovitskii and Milyutin (1965), and the necessary second-order optimality condition for bang-bang controls, see Milyutin and Osmolovskii (1998).

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Extremals in the problem of minimum time obstacle avoidance for a 2D double integrator system

Osmolovskii N. P., Figura A., Koska M., Wojtowicz M.

Control and Cybernetics

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2015

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Vol. 44, no. 2

185--209

EN

This paper provides an analysis of time optimal control problem of motion of a material point in the plane outside the given circle, without friction. The point is controlled by a force whose absolute value is limited by one. The closure of exterior of the circle plays the role of the state constraint. The analysis of the problem is based on the minimum principle.

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Second order conditions in optimal control problems with mixed equality-type constraints on a variable time interval

Osmolovskii N. P.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1535-1556

EN

This paper provides an analysis of second-order necessary or sufficient optimality conditions of Pontryagin or bounded strong minima, for optimal control problems of ordinary differential equations, considered on a nonfixed time interval, with constraints on initial-final time-state as well as mixed state-control constraints of equality type satisfying condition of linear independence of gradients w.r.t. control.

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Equivalence of second order optimality conditions for bang-bang control problems. Part 2 : Proofs, variational derivatives and representations

Osmolovskii N. P., Maurer H.

Control and Cybernetics

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2007

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Vol. 36, no 1

5-45

EN

In Part 1 of this paper (Osmolovskii and Maurer, 2005), we have summarized the main results on the equivalence of two quadratic forms from which second order necessary and sufficient conditions can be derived for optimal bang-bang control problems. Here, in Part 2, we give detailed proofs and elaborate explicit relations between Lagrange multipliers and elements of the critical cones in both approaches. The main analysis concerns the derivation of formulas for the first and second order derivatives of trajectories with respect to variations of switching times, initial and final time and initial point. This leads to explicit representations of the second order derivatives of the Lagrangian for the induced optimization problem. Based on a suitable transformation, we obtain the elements of the Hessian of the Lagrangian in a form which involves only first order variations of the nominal trajectory. Finally, a careful regrouping of all terms allows us to find the desired equivalence of the two quadratic forms.

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Equivalence of second order optimality conditions for bang-bang control problems. Part 1: Main results

Osmolovskii N. P., Maurer H.

Control and Cybernetics

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2005

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Vol. 34, no 3

927-950

EN

Second order optimality conditions have been derived in the literature in two different forms. Osmolovskii (1988a, 1995, 2000, 2004) obtained second order necessary and sufficient conditions requiring that, a certain quadratic form be positive (semi)-definite on a critical cone. Agrachev, Stefani, Zezza (2002) first, reduced the bang-bang control problem to finite-dimensional optimization and then show that well-known sufficient optimality conditions for this optimization problem supplemented by the strict bang-bang property furnish sufficient conditions for the bang-bang control problem. In this paper, we establish the equivalence of both forms of sufficient conditions and give explicit relations between corresponding Lagrange multipliers and elements of critical cones. Part 1 summarizes the main results while detailed proofs will be given in Part 2.

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Second order optimality conditions for bang-bang control problems

Maurer H., Osmolovskii N. P.

Control and Cybernetics

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2003

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Vol. 32, no 3

555-584

EN

Second order necessary and sufficient optimality conditions for bang-bang control problems have been studied in Milyutin, Osmolovskii (1998). These conditions amount to testing the positive (semi-)definiteness of a quadratic form on a critical cone. The assumptions are appropriate for numerical verification only in some special cases. In this paper, we study various transformations of the quadratic form and the critical cone which will be tailored to different types of control problems in practice. In particular, by means of a solution to a linear matrix differential equation, the quadratic form can be converted to perfect squares. We demonstrate by three practical examples that the conditions obtained can be verified numerically.