We consider an optimal control problem on a given interval [0, T] whose trajectories must satisfy the state constraint g (t, x(t)) ≤ 0 a.e. Infinite-dimensional perturbations of this constraint …
We consider an optimal control problem on a given interval [0, T] whose trajectories must satisfy the state constraint g (t, x(t)) ≤ 0 a.e. Infinite-dimensional perturbations of this constraint give rise to a value function V, whose epigraph is a closed set containing sensitivity information, controllability and penalization results, and even necessary conditions for optimality.
We consider nonlinear discrete optimal control problems with variable end points and with inequality and equality type constraints on trajectories and control. We derive first- and second-order necessary optimality conditions …
We consider nonlinear discrete optimal control problems with variable end points and with inequality and equality type constraints on trajectories and control. We derive first- and second-order necessary optimality conditions that are meaningful without a priori normality assumptions.
In this paper, we prove some regularity results for the value function v of an optimal control problem with state constraints. In particular, we are interested in studying the semiconcavity …
In this paper, we prove some regularity results for the value function v of an optimal control problem with state constraints. In particular, we are interested in studying the semiconcavity of v.
We study optimality conditions for various types of control problems like the standard optimal control problem, optimal multiprocesses, problems with infinite horizon or the control of Volterra integral equations. To …
We study optimality conditions for various types of control problems like the standard optimal control problem, optimal multiprocesses, problems with infinite horizon or the control of Volterra integral equations. To derive necessary conditions the needle variation method of Ioffe & Tichomirov is the central tool. In the particular control problem with infinite horizon the question of a suitable setting arises. We propose the framework of continuous state trajectories converging at infinity. This requires a version of Riesz' representation theorem and the introduction of regular Borel measures on the extended real number line. The control of Volterra integral equations including an inner and outer time variable. Consequently, we deal with a two-dimensional time set. We extend the needle variation method of Ioffe & Tichomirov to this case. The obtained optimality conditions are demonstrated in illustrative examples.
The infinite horizon optimal control problem is interpreted as the limit of unconstained problems with an extra running cost. The results is based on the viscosity solution method for the …
The infinite horizon optimal control problem is interpreted as the limit of unconstained problems with an extra running cost. The results is based on the viscosity solution method for the corresponding Bellman equations.
We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations …
We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.
We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the Clarke tangent cone of the …
We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the Clarke tangent cone of the state constraint set is non-empty (this is the constraint qualification that we suggest here), then the necessary optimality conditions apply in the normal form. We establish normality results for (weak) local minimizers and global minimizers, employing two different approaches and invoking slightly diverse assumptions. More precisely, for the local minimizers result, the Lagrangian is supposed to be Lipschitz with respect to the state variable, and just lower semicontinuous in its third variable. On the other hand, the approach for the global minimizers result (which is simpler) requires the Lagrangian to be convex with respect to its third variable, but the Lipschitz constant of the Lagrangian with respect to the state variable might now depend on time.
We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the Clarke tangent cone of the …
We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the Clarke tangent cone of the state constraint set is non-empty (this is the constraint qualification that we suggest here), then the necessary optimality conditions apply in the normal form. We establish normality results for (weak) local minimizers and global minimizers, employing two different approaches and invoking slightly diverse assumptions. More precisely, for the local minimizers result, the Lagrangian is supposed to be Lipschitz with respect to the state variable, and just lower semicontinuous in its third variable. On the other hand, the approach for the global minimizers result (which is simpler) requires the Lagrangian to be convex with respect to its third variable, but the Lipschitz constant of the Lagrangian with respect to the state variable might now depend on time.
Necessary and sufficient conditions for optimality in optimal control problems with state space constraints are reviewed with emphasis on geometric aspects.
Necessary and sufficient conditions for optimality in optimal control problems with state space constraints are reviewed with emphasis on geometric aspects.
Optimal control of piecewise deterministic processes with state space constraint is studied. Under appropriate assumptions, it is shown that the optimal value function is the only viscosity solution on the …
Optimal control of piecewise deterministic processes with state space constraint is studied. Under appropriate assumptions, it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain. Finally, the uniform continuity of the value function is obtained under a condition on the deterministic drift.
The continuity of the Lagrange multiplier $\mu(t)$ from the maximum principle for control problems with state constraints is investigated. It is proved, under certain regularity assumptions, that the function $\mu(t)$ …
The continuity of the Lagrange multiplier $\mu(t)$ from the maximum principle for control problems with state constraints is investigated. It is proved, under certain regularity assumptions, that the function $\mu(t)$ is Hölder continuous with exponent 1/2. Under regularity conditions plus the strong Legendre condition, this function is Lipschitz continuous.
In the present work, optimal control problems with mixed constraints are investigated. A novel weakening of the conventional regularity assumptions on mixed constraints is introduced. A maximum principle is derived …
In the present work, optimal control problems with mixed constraints are investigated. A novel weakening of the conventional regularity assumptions on mixed constraints is introduced. A maximum principle is derived in which the maximum condition is of nonstandard type: the maximum is taken over the closure of the set of regular points, but not over the whole feasible set.
A new method to prove non-degenerate optimality conditions in optimal control problems with state constraints of inequality type is proposed. This method involves the second-order derivative of the state constraint …
A new method to prove non-degenerate optimality conditions in optimal control problems with state constraints of inequality type is proposed. This method involves the second-order derivative of the state constraint function and is based on the notion of the extended Hamilton-Pontryagin function and on the related optimality conditions.