Metric sub-regularity in optimal control of affine problems with free end state

2020 ◽  
Vol 26 ◽  
pp. 47 ◽  
Author(s):  
N.P. Osmolovskii ◽  
V.M. Veliov
Keyword(s):  
Optimal Control ◽  
Terminal State ◽  
Optimality System ◽  
Convex Compact ◽  
Compact Set ◽  
First Order ◽  
Terminal Time ◽  
General Sufficient Condition ◽  
Convex Compact Set ◽  
Small Additive

The paper investigates the property of Strong Metric sub-Regularity (SMsR) of the mapping representing the first order optimality system for a Lagrange-type optimal control problem which is affine with respect to the control. The terminal time is fixed, the terminal state is free, and the control values are restricted in a convex compact set U. The SMsR property is associated with a reference solution of the optimality system and ensures that small additive perturbations in the system result in solutions which are at distance to the reference one, at most proportional to the size of the perturbations. A general sufficient condition for SMsR is obtained for appropriate space settings and then specialized in the case of a polyhedral set U and purely bang-bang reference control. Sufficient second-order optimality conditions are obtained as a by-product of the analysis. Finally, the obtained results are utilized for error analysis of the Euler discretization scheme applied to affine problems.

scholarly journals Optimal Control of Diffusions with Hard Terminal State Restrictions

2012 ◽  
Vol 2012 ◽  
pp. 1-38
Author(s):  
Atle Seierstad
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Terminal State ◽  
The State ◽  
State Equations ◽  
Brownian Motions ◽  
Terminal Time ◽  
State Restrictions

A maximum principle is proved for certain problems of optimal control of diffusions where hard end constraints occur. The results apply to several dimensional problems, where some of the state equations involve Brownian motions, but not the equations corresponding to states being hard restricted at the terminal time.

scholarly journals Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry

2021 ◽  
Author(s):  
Andrei Ardentov ◽  
Lev Lokutsievskiy ◽  
Yuri Sachkov
Keyword(s):  
Optimal Control Problems ◽  
Convex Compact ◽  
Control Problems ◽  
Compact Set ◽  
Dimensional Control ◽  
Rolling Ball ◽  
Dynamic Motion ◽  
Convex Compact Set ◽  
Motion Problem ◽  
Arbitrary Convex

We consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set Ω. The considered problems are well studied for the case when Ω is a unit disc, but barely studied for arbitrary Ω. We derive extremals to these problems in general case by using machinery of convex trigonometry, which allows us to do this identically and independently on the shape of Ω. The paper describes geodesics in (i)  the Finsler problem on the Lobachevsky hyperbolic plane; (ii) left-invariant sub-Finsler  problems on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with distance function given by Ω;  (iv) a series of "yacht problems" generalizing Euler's elastic problem, Markov-Dubins problem, Reeds-Shepp problem and a new sub-Riemannian problem on SE(2); and (v)  the plane dynamic motion problem.

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