Dante Kalise

Dante Kalise, Karl Kunisch

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the Successive Galerkin Approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear Generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.

Daisuke Inoue, Mathieu Laurière, Dante Kalise

The Schrödinger Bridge Problem constructs a stochastic process that connects an initial distribution to a terminal distribution with minimum energy. This work considers its mean-field extension, the Mean-Field Schrödinger Bridge, for interacting particle systems. With nonlocal interactions, evaluating the resulting particle-dependent distributional terms can scale quadratically with the population size, which makes large-scale problems intractable. We address this bottleneck by approximating the nonlocal interactions with neural network surrogates. The resulting four-stage alternating algorithm reduces the per-step cost from quadratic to linear in the population size at inference. We also derive Grönwall-type stability bounds that show how surrogate errors propagate to the generated trajectories. In numerical experiments on navigation and opinion-dynamics tasks, the proposed method reproduces trajectories obtained with analytical evaluation and reduces training time.

Rafael Bailo, Mattia Bongini, José A. Carrillo et al.

We study the numerical realisation of optimal consensus control laws for agent-based models. For a nonlinear multi-agent system of Cucker-Smale type, consensus control is cast as a dynamic optimisation problem for which we derive first-order necessary optimality conditions. In the case of a smooth penalization fo the control energy, the optimality system is numerically approximated via a gradient-descent method. For sparsity promoting, non-smooth $\ell_1$-norm control penalizations, the optimal controllers are realised by means of heuristic methods. For an increasing number of agents, we discuss the approximation of the consensus control problem by following a mean-field modelling approach.

Dante Kalise, Axel Kröner, Karl Kunisch

The numerical realization of the dynamic programming principle for continuous-time optimal control leads to nonlinear Hamilton-Jacobi-Bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls. This minimization is often performed by comparison over a finite number of elements of the control set. In this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively. The considered class of equations includes nonsmooth control problems with $\ell_1$-penalization which lead to sparse controls.

Luis Briceño-Arias, Dante Kalise, Ziad Kobeissi et al.

We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [SIAM J. Numer. Anal., 48(3):1136-1162, 2010]. In order to solve the finite dimensional variational problems, in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [J. Math. Imaging Vision, 40(1):120-145, 2011], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] by suitable preconditioned iterative algorithms.

Dante Kalise, Karl Kunisch, Zhiping Rao

A class of infinite horizon optimal control problems involving mixed quasi-norms of $L^p$-type cost functionals for the controls is discussed. These functionals enhance sparsity and switching properties of the optimal controls. The existence of optimal controls and their structural properties are analyzed on the basis of first order optimality conditions. A dynamic programming approach is used for numerical realization.

Dante Kalise, Lucas M. Moschen, Grigorios A. Pavliotis

We develop a feedback control framework for stabilizing the McKean-Vlasov PDE on the torus. Our goal is to steer the dynamics toward a prescribed stationary distribution or accelerate convergence to it using a time-dependent control potential. We reformulate the controlled PDE in a weighted, zero-mean space and apply the ground-state transform to obtain a Schrodinger-type operator. The resulting operator framework enables spectral analysis, verification of the infinite-dimensional Hautus test, and construction of a Riccati-based feedback law derived from the linearized dynamics, yielding local exponential stabilization with a chosen convergence rate. We rigorously prove local exponential stabilization via maximal regularity arguments and nonlinear estimates. Numerical experiments on well-studied models in one and two dimensions (the noisy Kuramoto model for synchronization, the O(2) spin model in a magnetic field, and the von Mises attractive interaction potential) showcase the effectiveness of our control strategy, demonstrating convergence acceleration and stabilization of unstable equilibria.

Dante Kalise, Ivar Lie, Eleuterio F. Toro

We present a numerical scheme for the solution of a class of atmospheric models where high horizontal resolution is required while a coarser vertical structure is allowed. The proposed scheme considers a layering procedure for the original set of equations, and the use of high-order ADER finite volume schemes for the solution of the system of balance laws arising from the dimensional reduction procedure. We present several types of layering based upon Galerkin discretizations of the vertical structure, and we study the effect of incrementing the order of horizontal approximation. Numerical experiments for the computational validation of the convergence of the scheme together with the study of physical phenomena are performed over 2D linear advective models, including a set of equations for an isothermal atmosphere.

M. Sajjad Edalatzadeh, Dante Kalise, Kirsten A. Morris et al.

Optimal actuator design for a vibration control problem is calculated. The actuator shape is optimized according to the closed-loop performance of the resulting linear-quadratic regulator and a penalty on the actuator size. The optimal actuator shape is found by means of shape calculus and a topological derivative of the linear-quadratic regulator (LQR) performance index. An abstract framework is proposed based on the theory for infinite-dimensional optimization of both the actuator shape and the associated control problem. A numerical realization of the optimality condition is presented for the actuator shape using a level-set method for topological derivatives. A Numerical example illustrating the design of actuator for Euler-Bernoulli beam model is provided.

Giacomo Albi, Dante Kalise

We study a multiscale approach for the control of agent-based, two-population models. The control variable acts over one population of leaders, which influence the population of followers via the coupling generated by their interaction. We cast a quadratic optimal control problem for the large-scale microscale model, which is approximated via a Boltzmann approach. By sampling solutions of the optimal control problem associated to binary two-population dynamics, we generate sub-optimal control laws for the kinetic limit of the multi-population model. We present numerical experiments related to opinion dynamics assessing the performance of the proposed control design.

Luis M. Briceño-Arias, José A. Carrillo, Dante Kalise et al.

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the $p$-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by computing explicit formulas for proximal operators with general costs amenable to efficient numerical approximation. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.

Carlos Doebeli, Alessandro Astolfi, Dante Kalise et al.

We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed to find an approximate solution to the invariance equation using a Newton iteration on the coefficients of a monomial basis expansion of the solution. These solutions to the invariance equations can then be used to construct reduced-order models. We assess the ability of the method to solve the invariance PDE system as well as to achieve moment matching and recover the steady-state behaviour of nonlinear systems with state dimension of order 1000 driven by linear and nonlinear signal generators.

Dante Kalise, Estefanía Loayza-Romero, Kirsten A. Morris et al.

Optimal actuator and control design is studied as a multi-level optimisation problem, where the actuator design is evaluated based on the performance of the associated optimal closed loop. The evaluation of the optimal closed loop for a given actuator realisation is a computationally demanding task, for which the use of a neural network surrogate is proposed. The use of neural network surrogates to replace the lower level of the optimisation hierarchy enables the use of fast gradient-based and gradient-free consensus-based optimisation methods to determine the optimal actuator design. The effectiveness of the proposed surrogate models and optimisation methods is assessed in a test related to optimal actuator location for heat control.

Giacomo Albi, Sara Bicego, Michael Herty et al.

Feedback control synthesis for large-scale particle systems is reviewed in the framework of model predictive control (MPC). The high-dimensional character of collective dynamics hampers the performance of traditional MPC algorithms based on fast online dynamic optimization at every time step. Two alternatives to MPC are proposed. First, the use of supervised learning techniques for the offline approximation of optimal feedback laws is discussed. Then, a procedure based on sequential linearization of the dynamics based on macroscopic quantities of the particle ensemble is reviewed. Both approaches circumvent the online solution of optimal control problems enabling fast, real-time, feedback synthesis for large-scale particle systems. Numerical experiments assess the performance of the proposed algorithms.

Sara Bicego, Samuel Gue, Dante Kalise et al.

A computational method for the synthesis of time-optimal feedback control laws for linear nilpotent systems is proposed. The method is based on the use of the bang-bang theorem, which leads to a characterization of the time-optimal trajectory as a parameter-dependent polynomial system for the control switching sequence. A deflated Newton's method is then applied to exhaust all the real roots of the polynomial system. The root-finding procedure is informed by the Hermite quadratic form, which provides a sharp estimate on the number of real roots to be found. In the second part of the paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network -- interpreted as a binary classifier -- via supervised learning. Numerical tests in integrators of increasing dimension assess the accuracy, robustness, and real-time-control capabilities of the approximate control law.

Giacomo Albi, Sara Bicego, Dante Kalise

A supervised learning approach for the solution of large-scale nonlinear stabilization problems is presented. A stabilizing feedback law is trained from a dataset generated from State-dependent Riccati Equation solves. The training phase is enriched by the use gradient information in the loss function, which is weighted through the use of hyperparameters. High-dimensional nonlinear stabilization tests demonstrate that real-time sequential large-scale Algebraic Riccati Equation solves can be substituted by a suitably trained feedforward neural network.

Dante Kalise, Sudeep Kundu, Karl Kunisch

We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reduced-order model, we construct a robust feedback control based on the $\cH_{\infty}$ control method, which requires the solution of an associated high-dimensional Hamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is tackled by means of a separable representation of the control system, and a polynomial approximation ansatz for the corresponding value function. Our method proves to be effective for the robust stabilization of nonlinear dynamics up to dimension $d\approx 12$. We assess the robustness and optimality features of our design over a class of nonlinear parabolic PDEs, including nonlinear advection and reaction terms. The proposed design yields a feedback controller achieving optimal stabilization and disturbance rejection properties, along with providing a modelling framework for the robust control of PDEs under parametric uncertainties.