This paper presents a novel motion planning approach inspired by the Dynamic Programming (DP) applicable to multi degree of freedom robots (mobile or stationary) and autonomous vehicles. The proposed discrete–time algorithm enables a robot to reach its destination through an arbitrary obstacle field in the fewest number of time steps possible while minimizing a secondary objective function. Furthermore, the resulting optimal trajectory is guaranteed to be globally optimal while incorporating state constraints such as velocity, acceleration, and jerk limits. The optimal trajectories furnished by the algorithm may be further updated in real time to accommodate changes in the obstacle field and/or cost function. The algorithm is proven to terminate in a finite number of steps without its computational complexity increasing with the type or number of obstacles. The effectiveness of the global and replanning algorithms are demonstrated on a planar mobile robot with three degrees of freedom subject to velocity and acceleration limits. The computational complexity of the two algorithms are also compared to that of an A*–type search.

In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error in L2(Ω) of the control variable is h | log h | in dimensions 2 and 3.

We discuss the zero-controllability and the zero-stabilizability for the nonnegative solutions to some Fisher-like models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. A necessary and sufficient condition for the nonnegative zero-stabilizabiity for a linear integro-partial differential equation is obtained in terms of the sign of the principal eigenvalue to a certain non-selfadjoint operator. For a related semilinear problem a necessary condition and a sufficient condition for the local nonnegative zero-stabilizability are also derived in terms of the magnitude of the above mentioned principal eigenvalue. The rate of stabilization corresponding to a simple feedback stabilizing control is dictated by the principal eigenvalue. A large principal eigenvalue leads to a fast stabilization to zero. A necessary condition and a sufficient condition for the stabilization to zero of the predator population in a predator-prey system is also investigated. Finally, a method to approximate the above mentioned principal eigenvalues is indicated.

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.

Sensitivity analysis (with respect to the regularization parameter)of the solution of a class of regularized state constrainedoptimal control problems is performed. The theoretical results arethen used to establish an extrapolation-based numerical scheme forsolving the regularized problem for vanishing regularizationparameter. In this context, the extrapolation technique providesexcellent initializations along the sequence of reducingregularization parameters. Finally, the favorable numericalbehavior of the new method is demonstrated and a comparison toclassical continuation methods is provided.

Necessary conditions of optimality in the form ofUnmaximized Inclusions (UI) are derived for optimal controlproblems with state constraints. The conditions presented heregeneralize earlier optimality conditions to problems that may benonconvex. The derivation of UI-type conditions in the absence of the convexity assumption is of particularimportance when deriving necessary conditions for constrainedproblems. We illustrate this feature by establishing, as anapplication, optimality conditions for problems that in additionto state constraints incorporate mixed state-control constraints.

The goal of this paper is to derive some error estimates for thenumerical discretization of some optimal control problems governedby semilinear elliptic equations with bound constraints on thecontrol and a finitely number of equality and inequality stateconstraints. We prove some error estimates for the optimalcontrols in the L ∞ norm and we also obtain error estimatesfor the Lagrange multipliers associated to the state constraintsas well as for the optimal states and optimal adjoint states.

The regularity of Lagrange multipliers for state-constrained optimal control problems belongs to the basic questions of controltheory. Here, we investigate bottleneck problems arising from optimal control problems for PDEs with certain mixed control-stateinequality constraints. We show how to obtain Lagrange multipliers in Lp spaces for linear problems and give an application to linearparabolic optimal control problems.