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We propose an optimization formulation for the simultaneous estimation of a latent variable and the identification of a linear continuous-time dynamic system, given a single input-output pair. We justify this approach based on Bayesian maximum a posteriori estimators. Our scheme takes the form of a convex alternating minimization, over the trajectories and the dynamic model respectively. We prove its convergence to a local minimum which verifies a two point-boundary problem for the (latent) state variable and a tensor product expression for the optimal dynamics.

A generic, fast and asymptotically efficient method for parametric estimation is described. It is based on the stochastic gradient descent on the loglikelihood function corrected by a single step of the Fisher scoring algorithm. We show theoretically and by simulations in the i.i.d. setting that it is an interesting alternative to the usual stochastic gradient descent with averaging or the adaptative stochastic gradient descent.

In this article, by using several new crucial {\it a priori} estimates which are still absent in the literature, we provide a comprehensive resolution of the first order generic mean field type control problems and also establish the global-in-time classical solutions of their Bellman and master equations. Rather than developing the analytical approach via tackling the Bellman and master equation directly, we apply the maximum principle approach by considering the induced forward-backward ordinary differential equation (FBODE) system; indeed, we first show the local-in-time unique existence of the solution of the FBODE system for a variety of terminal data by Banach fixed point argument, and then provide crucial a priori estimates of bounding the sensitivity of the terminal data for the backward equation by utilizing a monotonicity condition that can be deduced from the positive definiteness of the Schur complement of the Hessian matrix of the Lagrangian in the lifted version and manipulating first order condition appropriately; this uniform bound over the whole planning horizon $[0, T]$ allows us to partition $[0, T]$ into a number of sub-intervals with a common small length and then glue the consecutive local-in-time solutions together to form the unique global-in-time solution of the FBODE system. The regularity of the global-in-time solution follows from that of the local ones due to the regularity assumptions on the coefficient functions. Moreover, the regularity of the value function will also be shown with the aid of the regularity of the solution couple of the FBODE system and the regularity assumptions on the coefficient functions, with which we can further deduce that this value function and its linear functional derivative satisfy the Bellman and master equations, respectively.

In this article, we provide an original systematic global-in-time analysis of mean field type control problems on $\mathbb{R}^n$ with generic cost functionals by the modified approach but not the same, firstly proposed in [7], as the ``lifting'' idea introduced by P. L. Lions. As an alternative to the recent popular analytical method by tackling the master equation, we resolve the control problem in a certain proper Hilbert subspace of the whole space of $L^2$ random variables, it can be regarded as tangent space attached at the initial probability measure. The present work also fills the gap of the global-in-time solvability and extends the previous works of [7,11] which only dealt with quadratic cost functionals in control; the problem is linked to the global solvability of the Hilbert-space-valued forward-backward stochastic differential equation (FBSDE), which is solved by variational techniques here. We also rely on the Jacobian flow of the solution to this FBSDE to establish the regularities of the value function, including its linearly functional differentiability, which leads to the classical well-posedness of the Bellman equation. Together with the linear functional derivatives and the gradient of the linear functional derivatives of the solution to the FBSDE, we also obtain the classical well-posedness of the master equation.

International audience; It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories, allowing for a convenient handling of state constraints and meeting points. We now extend this viewpoint to estimation problems where it is known that kernels are the covariances of stochastic processes. Here, the Markovian Gaussian processes stem from the linear stochastic differential equations describing the continuous-time dynamics and observations. Taking extensive care to require minimal invertibility requirements on the operators, we give novel explicit formulas for these covariances. We also determine their reproducing kernel Hilbert spaces, stressing the symmetries between a space of forward-time trajectories and a space of backward-time information vectors. The two spaces play an analogue role for filtering to Sobolev spaces in variational analysis, and allow to recover the Kalman estimate through a direct variational argument. For comparison, we then recover the Kalman filter and smoother formulas through more classical arguments based on the innovation process. Extension to discrete-time observations or infinite-dimensional state, tough technical, would be straightforward.

International audience; The Linear Quadratic Regulator (LQR), which is arguably the most classical problem in control theory, was recently related to kernel methods in (Aubin-Frankowski, SICON, 2021) for finite dimensional systems. We show that this result extends to infinite dimensional systems, i.e.\ control of linear partial differential equations. The quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories, for which we obtain a concise formula based on the solution of the differential Riccati equation. This paves the way to applying representer theorems from kernel methods to solve infinite dimensional optimal control problems.

We extend the work \cite{bensoussan2019control} by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic and investigated a novel form of the `lifting' technique proposed by P. L. Lions. In \cite{bensoussan2019control}, we only showed the local existence and uniqueness of solutions to the FBODEs in the Hilbert space which were associated to the control problems with drift function consisting of the control only. In this article, we establish the global existence and uniqueness of the solutions to the FBODEs in Hilbert space corresponding to control problems with separable drift function which is nonlinear in state and linear in control. We shall also prove the sufficiency of the Pontryagin Maximum Principle and derive the corresponding Bellman equation. Besides, we shall show an analogue in the stationary case. Finally, by using the `lifting' idea as in \cite{stochasticv2,stochasticv1}, we shall apply the result to solve the linear quadratic mean field type control problems, and to show the global existence of the corresponding Bellman equations., Invited book chapter in Stochastic Analysis, Filtering, and Stochastic Optimization: A Commemorative Volume to Honor Mark H. A. Davies' Contributions

We survey in this article the connections between Machine Learning and Control Theory. Control Theory provide useful concepts and tools for Machine Learning. Conversely Machine Learning can be used to solve large control problems. In the first part of the paper, we develop the connections between reinforcement learning and Markov Decision Processes, which are discrete time control problems. In the second part, we review the concept of supervised learning and the relation with static optimization. Deep learning which extends supervised learning, can be viewed as a control problem. In the third part, we present the links between stochastic gradient descent and mean-field theory. Conversely, in the fourth and fifth parts, we review machine learning approaches to stochastic control problems, and focus on the deterministic case, to explain, more easily, the numerical algorithms.

The topic of identification of dynamic systems, has been at the core of modern control , following the fundamental works of Kalman. Realization Theory has been one of the major outcomes in this domain, with the possibility of identifying a dynamic system from an input-output relationship. The recent development of machine learning concepts has rejuvanated interest for identification. In this paper, we review briefly the results of realization theory, and develop some methods inspired by Machine Learning concepts.

We propose a new approach to studying classical solutions of the Bellman equation and Master equation for mean field type control problems, using a novel form of the "lifting" idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifiting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic.

The classical stochastic control problem under partial information can be formulated as a control problem for Zakai equation, whose solution is the unnormalized conditional probability distribution of the state of the system. Zakai equation is a stochastic Fokker-Planck equation. Therefore, the problem to be solved is similar to that met in Mean Field Control theory. Since Mean Field Control theory is much posterior to the development of Stochastic Control with partial information, the tools, techniques, and concepts obtained in the last decade, for Mean Field Games and Mean field type Control theory, have not been used for the control of Zakai equation. Our objective is to connect the two theories. We get the power of new tools, and we get new insights for the problem of stochastic control with partial information. For mean field theory, we get new interesting applications, but also new problems. Indeed, Mean Field Control Theory leads to very complex equations, like the Master equation, which is a nonlinear infinite dimensional P.D.E., for which general theorems are hardly available, although active research in this direction is performed. Direct methods are useful to obtain regularity results. We will develop in detail the LQ regulator problem, but since we cannot just consider the Gaussian case, well-known results, such as the separation principle is not available. An important result is available in the literature, due to A. Makowsky. It describes the solution of Zakai equation for linear systems with general initial condition (non-gaussian). We show that the separation principle can be extended for quadratic pay-off functionals, but the Kalman filter is much more complex than in the gaussian case. Finally we compare our work to the work of E. Bandini et al. and we show that the example E. Bandini et al. provided does not cover ours. Our system remains nonlinear in their setting., 29 pages

By extending \cite{bensoussan2015control}, we implement the proposal of Lions \cite{lions14} on studying mean field games and their master equations via certain control problems on the Hilbert space of square integrable random variables. In \cite{bensoussan2015control}, the Hilbert space could be quite general in the face of the "deterministic control problem" due to the absence of additional randomness; while the special case of $L^2$ space of square integrable random variables was brought in at the interpretation stage. The effectiveness of the approach was demonstrated by deriving Bellman equations and the first order master equations through control theory of dynamical systems valued in the Hilbert space. In our present problem for second order master equations, it connects with a stochastic control problem over the space of random variables, and it possesses an additional randomness generated by the Wiener process which cannot be detached from the randomness caused by the elements in the Hilbert space. Nevertheless, we demonstrate how to tackle this difficulty, while preserving most of the efficiency of the approach suggested by Lions \cite{lions14}., 28 pages main body, 9 pages appendix

In this paper, we investigate the interaction of two populations with a large number of indistinguishable agents. The problem consists in two levels: the interaction between agents of a same population, and the interaction between the two populations. In the spirit of mean field type control (MFC) problems and mean field games (MFG), each population is approximated by a continuum of infinitesimal agents. We define four different problems in a general context and interpret them in the framework of MFC or MFG. By calculus of variations, we derive formally in each case the adjoint equations for the necessary conditions of optimality. Importantly, we find that in the case of a competition between two coalitions, one needs to rely on a system of Master equations in order to describe the equilibrium. Examples are provided, in particular linear-quadratic models for which we obtain systems of ODEs that can be related to Riccati equations.

We consider a version of the stochastic inventory control problem for a spectrally positive L��vy demand process, in which the inventory can only be replenished at independent exponential times. We show the optimality of a periodic barrier replenishment policy that restocks any shortage below a certain threshold at each replenishment opportunity. The optimal policies and value functions are concisely written in terms of the scale functions. Numerical results are also provided., 27 pages, 3 figures. Forthcoming in SIAM Journal on Control and Optimization

We deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that we allow heavily mean field dependent dynamics. This in particular leads to a system of PDE's with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, we introduce a structural assumptions that cover many cases in stochastic differential games with mean filed dependent dynamics for which we are able to establish the existence of a weak solution. In addition, we present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis.

We study a system of partial differential equations used to describe Bertrand and Cournot competition among a continuum of producers of an exhaustible resource. By deriving new a priori estimates, we prove the existence of classical solutions under general assumptions on the data. Moreover, under an additional hypothesis we prove uniqueness. Keywords: mean field games, Hamilton-Jacobi, Fokker-Planck, coupled systems, optimal control, nonlinear partial differential equations

Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see [9], and also the very helpful notes of Cardialaguet [4] on Lions' lectures, the Master Equation has attracted a lot of interest, and various points of view have been expressed, see for example Carmona-Delarue [5], Bensoussan-Frehse-Yam [2], Buckdahn-Li-Peng-Rainer [3]. There are several ways to introduce this type of equation; and in those mentioned works, they involve an argument which is a probability measure, while P.L. Lions has recently proposed the idea of working with the Hilbert space of square integrable random variables. Hence writing the equation is an issue; while another issue is its origin. In this article, we discuss all these various aspects, and our modeling argument relies heavily on a seminar at College de France delivered by P.L. Lions on November 14, 2014.

We study in this paper a control problem in a space of random variables. We show that its Hamilton Jacobi Bellman equation is related to the Master equation in Mean field theory. P.L. Lions in [14,15] introduced the Hilbert space of square integrable random variables as a natural space for writing the Master equation which appears in the mean field theory. W. Gangbo and A. \'Swi\k{e}ch [10] considered this type of equation in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient. We compare the two approaches and provide some extension of the results of Gangbo and \'Swi\k{e}ch.

International audience; Earlier works in engineering, partly experimental, partly computational have revealed that asymptotically, when the excitation is a white noise, plastic deformation and total deformation for an elasto-perfectly-plastic oscillator have a variance which increases linearly with time with the same coefficient. In this work, we prove this result and we characterize the corresponding drift coefficient. Our study relies on a stochastic variational inequality governing the evolution between the velocity of the oscillator and the non-linear restoring force. We then define long cycles behavior of the Markov process solution of the stochastic variational inequality which is the key concept to obtain the result. An important question in engineering is to compute this coefficient. Also, we provide numerical simulations which show successful agreement with our theoretical prediction and previous empirical studies made by engineers.