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The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions...

Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints

J. Frédéric Bonnans, Audrey Hermant (2009)

Annales de l'I.H.P. Analyse non linéaire

Second-order necessary conditions for discrete inclusions with end point constraints

Aurelian Cernea (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We study an optimization problem given by a discrete inclusion with end point constraints. An approach concerning second-order optimality conditions is proposed.

Second-order sufficient condition for $\tilde{ℓ}$ -stable functions

Dušan Bednařík, Karel Pastor (2007)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The aim of our article is to present a proof of the existence of local minimizer in the classical optimality problem without constraints under weaker assumptions in comparisons with common statements of the result. In addition we will provide rather elementary and self-contained proof of that result.

Second-order sufficient conditions for strong solutions to optimal control problems

J. Frédéric Bonnans, Xavier Dupuis, Laurent Pfeiffer (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this article, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory...

Second-order sufficient optimality conditions for a semilinear optimal control problem with nonlocal radiation interface conditions

Christian Meyer (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, second-order sufficient conditions are derived that account for strongly active sets. These conditions ensure local optimality in an Ls-neighborhood of a reference function whereby the underlying analysis allows...

Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints

Nikolai P. Osmolovskii (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth...

Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints

Nikolai P. Osmolovskii (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations

Fredi Tröltzsch, Daniel Wachsmuth (2006)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper sufficient optimality conditions are established for optimal control of both steady-state and instationary Navier-Stokes equations. The second-order condition requires coercivity of the Lagrange function on a suitable subspace together with first-order necessary conditions. It ensures local optimality of a reference function in a $L^{s}$ -neighborhood, whereby the underlying analysis allows to use weaker norms than $L^{\infty}$ .

Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations

Fredi Tröltzsch, Daniel Wachsmuth (2005)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper sufficient optimality conditions are established for optimal control of both steady-state and instationary Navier-Stokes equations. The second-order condition requires coercivity of the Lagrange function on a suitable subspace together with first-order necessary conditions. It ensures local optimality of a reference function in a Ls-neighborhood, whereby the underlying analysis allows to use weaker norms than L∞.

Semicontinuità inferiore di funzionali integrali nel caso vettoriale e buona posizione nel calcolo delle variazioni

Silvia Bertirotti (2000)

Bollettino dell'Unione Matematica Italiana

Semi-group methods in stochastic control

A. Bensoussan (1985)

Banach Center Publications

Semi-smooth Newton methods for the Signorini problem

Kazufumi Ito, Karl Kunisch (2008)

Applications of Mathematics

Semi-smooth Newton methods are analyzed for the Signorini problem. A proper regularization is introduced which guarantees that the semi-smooth Newton method is superlinearly convergent for each regularized problem. Utilizing a shift motivated by an augmented Lagrangian framework, to the regularization term, the solution to each regularized problem is feasible. Convergence of the regularized problems is shown and a report on numerical experiments is given.

Semistochastic decomposition scheme in mathematical programming and game theory

Tran Quoc Chien (1991)

Kybernetika

Sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions.

Verma, Ram U. (2006)

Journal of Applied Mathematics and Stochastic Analysis

Sensitivity analysis of a nonlinear obstacle plate problem

Isabel N. Figueiredo, Carlos F. Leal (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9, 10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that...

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