On the Necessity of a Certain Convexity Condition for Lower Closure of Control Problems

1968 ◽  
Vol 6 (2) ◽  
pp. 174-185 ◽  
Author(s):  
Pavol Brunovský
Keyword(s):  
Control Problems ◽  
Convexity Condition ◽  
Lower Closure

scholarly journals Generic existence of solutions of nonconvex optimal control problems

2005 ◽  
Vol 2005 (4) ◽  
pp. 375-421 ◽  
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Optimal Control ◽  
Existence And Uniqueness ◽  
Optimal Control Problems ◽  
Existence Of Solutions ◽  
Integral Functional ◽  
Growth Conditions ◽  
Uniqueness Result ◽  
Control Problems ◽  
Convexity Condition ◽  
Generic Existence

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrandsfwhich satisfy convexity and growth conditions. In 1996, the author obtained a generic existence and uniqueness result (with respect to variations of the integrand of the integral functional) without the convexity condition for a class of optimal control problems satisfying the Cesari growth condition. In this paper, we survey this result and its recent extensions, and establish several new results in this direction.

Strong Lagrange duality and the maximum principle for nonlinear discrete time optimal control problems

2019 ◽  
Vol 25 ◽  
pp. 20
Author(s):  
Eero V. Tamminen
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Discrete Time ◽  
Optimal Control Problems ◽  
Discrete Maximum Principle ◽  
Control Problems ◽  
Convexity Condition ◽  
Lagrange Duality ◽  
The Maximum Principle ◽  
Time Optimal

We examine discrete-time optimal control problems with general, possibly non-linear or non-smooth dynamic equations, and state-control inequality and equality constraints. A new generalized convexity condition for the dynamics and constraints is defined, and it is proved that this property, together with a constraint qualification constitute sufficient conditions for the strong Lagrange duality result and saddle-point optimality conditions for the problem. The discrete maximum principle of Pontryagin is obtained in a straightforward manner from the strong Lagrange duality theorem, first in a new form in which the Lagrangian is minimized both with respect to the state and to the control variables. Assuming differentiability, the maximum principle is obtained in the usual form. It is shown that dynamic systems satisfying a global controllability condition with convex costs, have the required convexity property. This controllability condition is a natural extension of the customary directional convexity condition applied in the derivation of the discrete maximum principle for local optima in the literature.

Self-Control Problems Measure

2011 ◽  
Author(s):  
Kevin M. King ◽  
Charles B. Fleming ◽  
Kathryn C. Monahan ◽  
Richard F. Catalano
Keyword(s):  
Self Control ◽  
Control Problems

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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