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Minima in control problems with constraints

Gianna Stefani, PierLuigi Zezza (1995)

Banach Center Publications

This paper is devoted to describing second order conditions in the framework of extremal problems, that is, conditions obtained by reducing the optimal control problem to an abstract one in a suitable Banach (or Hilbert) space. The studied problem includes equality constraints both on the end-points and on the state-control trajectory. The second goal is to give a complete description of necessary and sufficient second order conditions for weak local optimality by describing first the associated...

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.


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

Tadeusz Kaczorek (2014)

International Journal of Applied Mathematics and Computer Science

A minimum energy control problem for fractional 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.

Monotonicity properties of minimizers and relaxation for autonomous variational problems

Giovanni Cupini, Cristina Marcelli (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the following classical autonomous variational problem minimize F ( v ) = a b f ( v ( x ) , v ' ( x ) ) x ̣ : v A C ( [ a , b ] ) , v ( a ) = α , v ( b ) = β , where the Lagrangianf is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

Monotonicity properties of minimizers and relaxation for autonomous variational problems

Giovanni Cupini, Cristina Marcelli (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the following classical autonomous variational problem minimize F ( v ) = a b f ( v ( x ) , v ' ( x ) ) x ̣ : v A C ( [ a , b ] ) , v ( a ) = α , v ( b ) = β , where the Lagrangian f is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

Necessary Optimality Conditions for a Lotka-Volterra Three Species System

N. C. Apreutesei (2010)

Mathematical Modelling of Natural Phenomena

An optimal control problem is studied for a Lotka-Volterra system of three differential equations. It models an ecosystem of three species which coexist. The species are supposed to be separated from each others. Mathematically, this is modeled with the aid of two control variables. Some necessary conditions of optimality are found in order to maximize the total number of individuals at the end of a given time interval.

Nonnegativity of functionals corresponding to the second order half-linear differential equation

Robert Mařík (1999)

Archivum Mathematicum

In this paper we study extremal properties of functional associated with the half–linear second order differential equation E p . Necessary and sufficient condition for nonnegativity of this functional is given in two special cases: the first case is when both points are regular and the second is the case, when one end point is singular. The obtained results extend the theory of quadratic functionals.

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