scholarly journals Singular extremals in L1-optimal control problems: sufficient optimality conditions

2020 ◽  
Vol 26 ◽  
pp. 99
Author(s):  
Francesca C. Chittaro ◽  
Laura Poggiolini
Keyword(s):  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Sufficient Optimality Conditions ◽  
Local Optimality ◽  
Second Variation ◽  
Absolute Value ◽  
Singular Extremals ◽  
Hamiltonian Methods ◽  
The Second Variation

In this paper we are concerned with generalised L1-minimisation problems, i.e. Bolza problems involving the absolute value of the control with a control-affine dynamics. We establish sufficient conditions for the strong local optimality of extremals given by the concatenation of bang, singular and inactive (zero) arcs. The sufficiency of such conditions is proved by means of Hamiltonian methods. As a by-product of the result, we provide an explicit invariant formula for the second variation along the singular arc.

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.

Geometric constrained variational calculus. II: The second variation (Part I)

2016 ◽  
Vol 13 (01) ◽  
pp. 1550132 ◽  
Author(s):  
Enrico Massa ◽  
Danilo Bruno ◽  
Gianvittorio Luria ◽  
Enrico Pagani
Keyword(s):  
Gauge Transformation ◽  
Sufficient Conditions ◽  
Variational Calculus ◽  
Second Variation ◽  
Action Functional ◽  
Covariant Representation ◽  
Jacobi Fields ◽  
Necessary And Sufficient ◽  
The Second Variation ◽  
Variation Part

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

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