scholarly journals Pontryagin Maximum Principle for Finite Dimensional Nonlinear Optimal Control Problems on Time Scales

2013 ◽  
Vol 51 (5) ◽  
pp. 3781-3813 ◽  
Author(s):  
Loïc Bourdin ◽  
Emmanuel Trélat
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Time Scales ◽  
Pontryagin Maximum Principle ◽  
Optimal Control Problems ◽  
Nonlinear Optimal Control ◽  
Control Problems ◽  
Finite Dimensional

Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity

2018 ◽  
Vol 24 (4) ◽  
pp. 1705-1734 ◽  
Author(s):  
Roman Šimon Hilscher ◽  
Vera Zeidan
Keyword(s):  
Optimal Control ◽  
Time Scales ◽  
Shooting Method ◽  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Nonlinear Optimal Control ◽  
Control Problems ◽  
Local Minimality ◽  
The Second Variation ◽  
Accessory Problem

The main focus of this paper is to develop a sufficiency criterion for optimality in nonlinear optimal control problems defined on time scales. In particular, it is shown that the coercivity of the second variation together with the controllability of the linearized dynamic system are sufficient for the weak local minimality. The method employed is based on a direct approach using the structure of this optimal control problem. The second aim pertains to the sensitivity analysis for parametric control problems defined on time scales with separately varying state endpoints. Assuming a slight strengthening of the sufficiency criterion at a base value of the parameter, the perturbed problem is shown to have a weak local minimum and the corresponding multipliers are shown to be continuously differentiable with respect to the parameter. A link is established between (i) a modification of the shooting method for solving the associated boundary value problem, and (ii) the sufficient conditions involving the coercivity of the accessory problem, as opposed to the Riccati equation, which is also used for this task. This link is new even for the continuous time setting.

scholarly journals A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problems

2019 ◽  
Vol 25 ◽  
pp. 52 ◽  
Author(s):  
Benoît Bonnet
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Pontryagin Maximum Principle ◽  
Optimal Control Problems ◽  
Classical Method ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
Non Local ◽  
Wasserstein Space ◽  
Needle Variations

In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics is described by a transport equation with non-local velocities which are affine in the control, and is subject to end-point and running state constraints. Building on our previous work, we combine the classical method of needle-variations from geometric control theory and the metric differential structure of the Wasserstein spaces to obtain a maximum principle formulated in the so-called Gamkrelidze form.

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