Second-order sensitivity relations and regularity of the value function for Mayer's problem in optimal control

This paper investigates the value function, $V$, of a Mayer optimal control problem with the state equation given by a differential inclusion. First, we obtain an invariance property for the proximal and Fr\'echet subdifferentials of $V$ along optimal trajectories. Then, we extend the analysis to the sub/superjets of $V$, obtaining new sensitivity relations of second order. By applying sensitivity analysis to exclude the presence of conjugate points, we deduce that the value function is twice differentiable along any optimal trajectory starting at a point at which $V$ is proximally subdifferentiable. We also provide sufficient conditions for the local $C^2$ regularity of $V$ on tubular neighborhoods of optimal trajectories

Regularity results for the minimum time function with H\"ormander vector fields

In a bounded domain of $\mathbb{R}^n$ with smooth boundary, we study the regularity of the viscosity solution, $T$, of the Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\ldots ,X_N\}$, subject to H\"ormander's bracket generating condition. Due to the presence of characteristic boundary points, singular trajectories may occur in this case. We characterize such trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. We then prove that the local Lipschitz continuity of $T$, the local semiconcavity of $T$, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied when the characteristic set of $\{X_1,\ldots ,X_N\}$ is a symplectic manifold. We apply our results to Heisenberg's and Martinet's vector fields

Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a $L^1$-penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation

A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. Measurability, local convergence, and convergence rates of a stationarity measure are handled, filling a gap for applications to nonconvex infinite dimensional stochastic optimization problems. The method is demonstrated on an optimal control problem constrained by a class of elliptic semilinear partial differential equations (PDEs) under uncertainty

Relations de sensibilité et régularité des solutions d'une classe d'équations d'HJB en controle optimal

Dans cette thèse nous étudions une classe d’équations de Hamilton-Jacobi-Bellman provenant de la théorie du contrôle optimal des équations différentielles ordinaires. Nous nous intéressons principalement à l’analyse de la sensibilité de la fonction valeur des problèmes de contrôle optimal associés à de telles équations de H-J-B. Dans la littérature, les relations de sensibilité fournissent une “mesure” de la robustesse des stratégies optimales par rapport aux variations de la variable d’état. Ces résultats sont des outils très importants pour le contrôle appliqué, parce qu’ils permettent d’étudier les effets que des approximations des données du système peuvent avoir sur les politiques optimales. Cette thèse est dédiée également à l’étude des problèmes de Mayer et de temps minimal. Nous supposons que la dynamique du problème soit une inclusion différentielle, afin de permettre aux données d’être non régulières et d’embrasser un ensemble plus grand d’applications. Néanmoins, cette tâche rend notre analyse plus difficile. La première contribution de cette étude est une extension de quelques résultats classiques de la théorie de la sensibilité au domaine des problèmes non paramétrées. Ces relations prennent la forme d’inclusions d’état adjoint, figurant dans le principe du maximum de Pontryagin, dans certains gradients généralisés de la fonction valeur évalués le long des trajectoires optimales. En deuxième lieu, nous développons des nouvelles relations de sensibilité impliquant des approximations du deuxième ordre de la fonction valeur. Cette analyse mène à de nouvelles applications concernant la propagation, tant ponctuel que local, de la régularité de la fonction valeur le long des trajectoires optimales. Nous proposons également des applications aux conditions d’optimalité.This dissertation investigates a class of Hamilton-Jacobi-Bellman equations arising in optimal control of O.D.E.. We mainly focus on the sensitivity analysis of the optimal value function associated with the underlying control problems. In the literature, sensitivity relations provide a measure of the robustness of optimal control strategies with respect to variations of the state variable. This is a central tool in applied control, since it allows to study the effects that approximations of the inputs of the system may produce on the optimal policies. In this thesis, we deal whit problems in the Mayer or in the minimum time form. We assume that the dynamic is described by a differential inclusion, in order to allow data to be nonsmooth and to embrace a large area of concrete applications. Nevertheless, this task makes our analysis more challenging. Our main contribution is twofold. We first extend some classical results on sensitivity analysis to the field of nonparameterized problems. These relations take the form of inclusions of the co-state, featuring in the Pontryagin maximum principle, into suitable gradients of the value function evaluated along optimal trajectories. Furthermore, we develop new second-order sensitivity relations involving suitable second order approximations of the optimal value function. Besides being of intrinsic interest, this analysis leads to new consequences regarding the propagation of both pointwise and local regularity of the optimal value functions along optimal trajectories. As applications, we also provide refined necessary optimality conditions for some class of differential inclusions

Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a L1-penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation

Regularity results for the minimum time function with H\uf6rmander vector fields

In a bounded domain of R-n with boundary given by a smooth (n - 1)-dimensional manifold, we consider the homogeneous Dirichlet problem for the eikonal equation associated with a family of smooth vector fields X-1,..., X-N subject to Hormander's bracket generating condition. We investigate the regularity of the viscosity solution Tof such problem. Due to the presence of characteristic boundary points, singular trajectories may occur. First, we characterize these trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. Then, we prove that the local Lipschitz continuity of T, the local semiconcavity of T, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied whenever the characteristic set of X-1,..., X-N is a symplectic manifold. We apply our results to several examples. (c) 2017 Elsevier Inc. All rights reserved

High order discrete approximations to Mayer's problems for linear systems

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Parental involvement in the care and intervention of children with hearing loss

Objective: The present study aimed to explore the nature of parental involvement in the intervention of children with hearing loss, as experienced by parents. Design: A qualitative descriptive methodology was adopted to conduct semi-structured in-depth interviews with a purposive sample of parents who have a child with hearing loss. Study sample: Seventeen parents of 11 children aged 6–9 years participated in this study. Results: The overarching theme of parents taking the central role was identified using thematic analysis. This overarching theme connected five themes which described the nature of parental involvement: (1) parents work behind the scenes; (2) parents act as ‘case managers’; (3) parents always have their child’s language development in mind; (4) parents’ role extends to advocacy for all children with hearing loss; and (5) parents serve a number of roles, but at the end of the day, they are parents. Conclusions: The results indicate that parental involvement in the intervention of children with hearing loss is multifaceted in nature and incorporates a broad range of behaviours and practices. These findings have important implications for the provision of family-centred practices

The parents’ perspective of the early diagnostic period of their child with hearing loss: information and support

This study aimed to explore the perspectives of caregivers regarding the information and support they received following diagnosis of their child's hearing loss.A mixed methods explanatory sequential design was conducted.A total of 445 caregivers of children completed a written survey, and five parents participated in qualitative in-depth interviews.The most common sources of information for caregivers were discussion with an audiologist, written information, and discussion with a medical professional. Approximately 85% of caregivers reported they were satisfied with the personal/emotional support and information received from service providers. Additional comments from 91 caregivers indicated that 11% experienced a breakdown in information transfer with health professionals. Interviews conducted with five parents from three families revealed two themes which described the diagnostic period as a difficult and emotional experience for parents: (1) support and information provided during diagnosis: what happens first? and (2) accessing early intervention services following a diagnosis of hearing loss: navigating the maze.The findings of this study give insight into the perspectives of caregivers who have a child diagnosed with hearing loss. The importance of providing timely information and personal/emotional support to caregivers cannot be underestimated