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Minimax optimal control problems. Numerical analysis of the finite horizon case

Silvia C. Di Marco, Roberto L.V. González (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we consider the numerical computation of the optimal cost function associated to the problem that consists in finding the minimum of the maximum of a scalar functional on a trajectory. We present an approximation method for the numerical solution which employs both discretization on time and on spatial variables. In this way, we obtain a fully discrete problem that has unique solution. We give an optimal estimate for the error between the approximated solution and the optimal cost function...

Minimax theorems with applications to convex metric spaces

Jürgen Kindler (1995)

Colloquium Mathematicae

A minimax theorem is proved which contains a recent result of Pinelis and a version of the classical minimax theorem of Ky Fan as special cases. Some applications to the theory of convex metric spaces (farthest points, rendez-vous value) are presented.

Minimax theorems without changeless proportion

Liang-Ju Chu, Chi-Nan Tsai (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: i n f y ∈ Y s u p x ∈ X f ( x , y ) ≤ s u p x ∈ X i n f y ∈ Y g ( x , y ) . We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: s u p x ∈ X f ( x , y ) ≤ s u p x ∈ X g ( x , y ) , ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties...

Minimization of functional with integrand expressed as minimum of quasiconvex functions - general and special cases

Piotr Puchała (2014)

Banach Center Publications

We present Z. Naniewicz method of optimization a coercive integral functional 𝒥 with integrand being a minimum of quasiconvex functions. This method is applied to the minimization of functional with integrand expressed as a minimum of two quadratic functions. This is done by approximating the original nonconvex problem by appropriate convex ones.

Minimizers with topological singularities in two dimensional elasticity

Xiaodong Yan, Jonathan Bevan (2008)

ESAIM: Control, Optimisation and Calculus of Variations

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of S 1 ; the minimizer u is C 1 and is such that det ∇ u vanishes at one point.

Minimizers with topological singularities in two dimensional elasticity

Jonathan Bevan, Xiaodong Yan (2010)

ESAIM: Control, Optimisation and Calculus of Variations

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of S1; the minimizer u is C1 and is such that det ∇ u vanishes at one point.


Minimizing p -harmonic maps at a free boundary

Frank Duzaar, Andreas Gastel (1998)

Bollettino dell'Unione Matematica Italiana

Studiamo le proprietà di regolarità delle mappe fra varietà di Riemann che minimizzano la p -energia fra quelle che soddisfano una condizione di frontiera pazialmente libera. Proviamo che tali mappe sono Hölder continue vicino alla frontiera libera fuori di un insieme singolare, e otteniamo stime ottimali per la dimensione di Hausdorff di questo insieme singolare.

Minimizing the fuel consumption of a vehicle from the Shell Eco-marathon: a numerical study

Sophie Jan (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We apply four different methods to study an intrinsically bang-bang optimal control problem. We study first a relaxed problem that we solve with a naive nonlinear programming approach. Since these preliminary results reveal singular arcs, we then use Pontryagin’s Minimum Principle and apply multiple indirect shooting methods combined with homotopy approach to obtain an accurate solution of the relaxed problem. Finally, in order to recover a purely bang-bang solution for the original problem, we...

Minimum energy control of positive continuous-time linear systems with bounded inputs

Tadeusz Kaczorek (2013)

International Journal of Applied Mathematics and Computer Science

The minimum energy control problem for positive continuous-time linear systems with bounded inputs is formulated and solved. Sufficient conditions for the existence of a solution to the problem are established. A procedure for solving the problem is proposed and illustrated with a numerical example.

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