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Pointwise minimization of supplemented variational problems

Peter Kosmol, Dieter Müller-Wichards (2004)

Colloquium Mathematicae

We describe an approach to variational problems, where the solutions appear as pointwise (finite-dimensional) minima for fixed t of the supplemented Lagrangian. The minimization is performed simultaneously with respect to the state variable x and ẋ, as opposed to Pontryagin's maximum principle, where optimization is done only with respect to the ẋ-variable. We use the idea of the equivalent problems of Carathéodory employing suitable (and simple) supplements to the original minimization problem....

Porosity and Variational Principles

Marchini, Elsa (2002)

Serdica Mathematical Journal

We prove that in some classes of optimization problems, like lower semicontinuous functions which are bounded from below, lower semi-continuous or continuous functions which are bounded below by a coercive function and quasi-convex continuous functions with the topology of the uniform convergence, the complement of the set of well-posed problems is σ-porous. These results are obtained as realization of a theorem extending a variational principle of Ioffe-Zaslavski.

Preface

Zbigniew Bartosiewicz, Ewa Girejko (2006)

Control and Cybernetics

Projective Reeds-Shepp car on S2 with quadratic cost

Ugo Boscain, Francesco Rossi (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Fix two points x , x ¯ ∈ S 2 and two directions (without orientation) η , η ¯ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost J [ γ ] = ∫ 0 T γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) + K γ ( t ) 2 γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t along all smooth curves starting from x with direction η and ending in x ¯ with direction η ¯ . Here g is the standard Riemannian metric on S2 and K γ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We...

Proper orthogonal decomposition for optimality systems

Karl Kunisch, Stefan Volkwein (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled...

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