Search

We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero. [ABSTRACT FROM PUBLISHER]

A classical problem in ergodic control consists of studying the limit behavior of the optimal value Vλ of a discounted cost functional with infinite horizon as the discount factor λ tends to zero. In the literature, this problem has been addressed under various conditions ensuring that the rescaled value function λVλ converges uniformly to a constant limit. The main goal of this paper is to study this problem without such conditions, so that the aforementioned limit need not be constant. So, under a nonexpansivity assumption, we derive Lipschitz bounds which yield compactness of {λVλ} for both control systems and differential games. Then, we study the convergence of solutions to Hamilton--Jacobi equations under the hypothesis that the Hamiltonian is radially nondecreasing, hence allowing for the existence noncoercivity directions. Using PDE methods, we show that the convergence is monotone and we characterize the limit as the maximal subsolution of a certain Hamilton--Jacobi equation. Finally, we return to optimal control problems and differential games for nonexpansive dynamics obtaining explicit representation formulas for the uniform limit of λVλ as λ → 0. [ABSTRACT FROM AUTHOR]

This paper studies stability properties of the solutions of optimal control problems for linear systems. The analysis is based on an adapted concept of metric regularity, the strong bi-metric regularity, which is introduced and investigated in the paper. It allows one to give a more precise description of the effect of perturbations on the optimal solutions in terms of a Hölder- type estimate and to investigate the robustness of this estimate. The Hölder exponent depends on a natural number k, which is known as the controllability index of the reference solution. An inverse function theorem for strongly bi-metrically regular mappings is obtained, which is used in the case k = 1 for proving stability of the solution of the considered optimal control problem under small nonlinear perturbations. Moreover, a new stability result with respect to perturbations in the matrices of the system is proved in the general case k ≥ 1. [ABSTRACT FROM AUTHOR]

Abstract: We give a representation for the closed convex hull of the set of discounted occupational measures generated by control-state trajectories of a deterministic control system. We also investigate the limit behavior of the latter when the discount factor tends to zero and compare it with the limit behavior of the long run time average occupational measures set. The novelty of our results is in that we allow the control set dependence on the state variables that make the results to be applicable to differential inclusions. [Copyright &y& Elsevier]

We study a zero-sum differential game where the players have only an unperfect information on the state of the system. In the beginning of the game only a random distribution on the initial state is available. The main result of the paper is the existence of the value obtained through an uniqueness result for Hamilton-Jacobi-Isaacs equations stated on the space of measure in ℝn. This result is the first step for future work on differential games with lack of information. [ABSTRACT FROM AUTHOR]

In the present paper, we prove that the optimal cost of a Mayer''s problem for a control system with singular perturbation does converge to the optimal cost of a Mayer''s problem associated with control system obtained by averaging. The main novelty of our result lies on the fact that we did not require any continuity assumption on the final cost. [Copyright &y& Elsevier]

Presents a study that aimed to give a representation formula for the mean curvature motion (MCM) in terms of the value function of some stochastic optimal control problem. Distance solution of the MCM; Discussion on the study of the distance solutions; Generalization of the results to general geometric equations.

Abstract. We characterize functions satisfying a dissipative inequality associated with a control problem. Such a characterization is provided in terms of an epicontingent solution, or a viscosity supersolution to a partial differential equation called Isaacs' equation. Links between supersolutions and epicontingent solutions to Isaacs' equation are studied. Finally, we derive (possibly discontinuous) disturbance attenuation feedback of the H[sup∞] problem from contingent formulation of Isaacs' equation. [ABSTRACT FROM AUTHOR]

This study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on R d representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton-Jacobi-Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton-Jacobi equations in the Wasserstein space., (© The Author(s) 2021.)

This paper considers a class of neural networks (NNs) for solving linear programming (LP) problems, convex quadratic programming (QP) problems, and nonconvex QP problems where an indefinite quadratic objective function is subject to a set of affine constraints. The NNs are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic, which enable to implement an exact penalty method. A new method is exploited to address convergence of trajectories, which is based on a nonsmooth Lojasiewicz inequality for the generalized gradient vector field describing the NN dynamics. The method permits to prove that each forward trajectory of the NN has finite length, and as a consequence it converges toward a singleton. Furthermore, by means of a quantitative evaluation of the Lojasiewicz exponent at the equilibrium points, the following results on convergence rate of trajectories are established: (1) for nonconvex QP problems, each trajectory is either exponentially convergent, or convergent in finite time, toward a singleton belonging to the set of constrained critical points; (2) for convex QP problems, the same result as in (1) holds; moreover, the singleton belongs to the set of global minimizers; and (3) for LP problems, each trajectory converges in finite time to a singleton belonging to the set of global minimizers. These results, which improve previous results obtained via the Lyapunov approach, are true independently of the nature of the set of equilibrium points, and in particular they hold even when the NN possesses infinitely many nonisolated equilibrium points.

Twenty patients with hypogonadism (19 women with amenorrhoea and 1 man with impotence and infertility), galactorrhoea and hyper-prolactinaemia (range: 36 to 344 ng/ml) were studied. The radiological study of the sella turcica, including in all cases hypocycloidal tomograms, allowed classification of the patients into 3 groups: group I (n = 4) had a grossly enlarged sella turcica, group II (n = 12) had localized alterations indicating the probable existence of a prolactin-secreting microadenoma ("microdeformation") while group III patients presented no radiological abnormality. Before treatment, all the patients were submitted to a complete evaluation of the function of their anterior pituitary, including the LH and FSH responses to iv administration of Gn-RH. All the group I patients had low basal LH levels and a blunted response to Gn-RH. The basal LH and in response to Gn-RH were normal in most of the group II patients and in all of the group III patients. An exaggerated FSH response to Gn-RH was observed in 6/12 patients with microdeformation (group II) but not in groups I and III patients. A low LH and a blunted LH response to Gn-RH is highly suggestive of the existence of a pituitary prolactin-secreting adenoma in case of amenorrhoea and hyper-prolactinaemia patients; a normal response does not however rule out such a diagnosis. The reasons for a exaggerated FSH response to Gn-RH in patients with suspected prolactin-secreting microadenoma remain to be investigated though this pattern can also occur in other cases of amenorrhoea. Hence the Gn-RH test might contribute to the assessment of the hypothalamo-pituitary axis of patients with hyper-prolactinaemia. Six patients treated for 4 months with bromocriptine (CB-154) were submitted to re-evaluation of their pituitary gonadotrophins reserve. All the women experienced restoration of menses with 39 days of treatment and the male patient regained potency. It was observed that bromocriptine treatment and subsequent normalized prolactin levels in the 4 group II women tested were associated with normalization of their previously exaggerated FSH response to Gn-RH; LH responses were also diminished in these cases. These data are compatible with the hypothesis that hyper-prolactinaemia per se could interfere with the endogenous secretion of Gn-RH at the hypothalamic level. In one patient with grossly enlarged sella turcica and a previous lack of an LH and FSH response to Gn-RH, bromocriptine treatment restored a normal gonadotrophins response, confirming that, in this case, the alteration of this response was indeed due to a prolonged lack of endogenous Gn-RH secretion.