# Day 2 - 16th of May

**09:30 – Bosco García-Archilla: Optimal bounds for POD approximations of infinite horizon control problems based on time derivatives**

We consider the numerical approximation of infinite horizon problems via the dynamic programming approach. The value function of the problem solves a Hamilton-Jacobi-Bellman (HJB) equation that is approximated by a fully discrete method. It is known that the numerical problem is difficult to handle by the so called curse of dimensionality. To mitigate this issue we apply a reduction of the order by means of a new proper orthogonal decomposition (POD) method based on time derivatives.

We carry out the error analysis of the method using recently proved optimal bounds for the fully discrete approximations. Moreover, the use of snapshots based on time derivatives allow us to bound some terms of the error that could not be bounded in a standard POD approach. Some numerical experiments show the good performance of the method in practice.

**10:00 – Julia Novo (& Javier de Frutos): Optimal bounds for numerical approximations of infinite horizon problems based on dynamic programming approach**

In this talk we show the error bounds that can be obtained for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering only the time discretization with a positive step size error bounds of size are obtained for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size an error bound of size can be found in the literature for the error in the fully discrete value function. In this talk, we revise the error bounds of the fully discrete method and prove that, under similar assumptions to those of the time discrete case, the error in the fully discrete case is which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behavior from the bound had never been observed.

**10:30 – Coffee break**

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**11:00 – Luca Saluzzi: Decaying sensitivity for the resolution of high-dimensional optimal control problems**

Optimal control problems arise in a variety of applications and one way to approach such problems is by computing the optimal value function by solving a Hamilton-Jacobi-Bellman (HJB)- PDE. However, the computational complexity of grid-based numerical methods for this problem grows exponentially with respect to the state space dimension, a phenomenon known as the curse of dimensionality.

To mitigate this problem, in this talk we describe the dynamical system as a set of subsystems interconnected via a graph. It seems reasonable to expect that subsystems that are far away (in terms of the graph distance) only interact with each other very weakly. Based on this decaying sensitivity assumption between subsystems, we are able to build a separable approximation of the optimal value function based on neighborhoods in the graph. Under sufficient regularity assumptions, separable functions can be approximated by deep neural networks with a number of neurons that grows only polynomially in the state dimension, leading to a mitigation of the curse of dimensionality. We will then focus on the LQR case, where it is possible to prove this decaying sensitivity by studying the corresponding Algebraic Riccati Equation and we will investigate numerical approximations able to exploit the sparsity structure.

**11:30 – Vincent Liu: A Hamilton-Jacobi-Bellman approach to ellipsoidal approximations of reachable sets for linear time-varying systems**

Society's ever-increasing integration of autonomous systems in day-to-day life has simultaneously brought forth concerns as to how their safety and reliability can be verified. To this end, reachable sets lend themselves well to this task. These sets describe collections of states that a dynamical system can reach in finite time, which can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse-of-dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this talk, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. This result arises from the construction of a local viscosity supersolution and subsolution of a Hamilton-Jacobi-Bellman equation for the corresponding reachability problem. The proposed ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making the approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.

**12:00 – Tobias Ehring: Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems**

Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context knowledge by its structure. Especially, the value function surrogate is enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm allows formulation in a matrix-free way which ensures efficient offline and online evaluation of the surrogate, circumventing the large-matrix problem for multivariate Hermite interpolation. Additionally, an incremental Cholesky factorization is utilized in the offline generation of the surrogate. For finite time horizons, both convergence of the surrogate to the value function and for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model with an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.

**12:30 – Lunch break**

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**14:30 – Maria Strazzullo: Regularized full and reduced feedback control strategies for convection-dominated Navier-Stokes equations**

In numerous scientific and industrial contexts, controlling the flow regime is crucial. There is a growing interest in the subject with the main goal of steering the flow towards beneficial configurations, less turbulent and more laminar. To address this, we propose a novel linear feedback control method for the Navier-Stokes equations with high Reynolds numbers, ensuring exponential convergence to desired state. However, as Reynolds numbers increase, additional stabilization strategies become necessary. We exploit the Evolve-Filter-Relax (EFR) algorithm, showing its theoretical non-exponential convergence. Guided by these insights, we present an adaptive EFR (aEFR) approach, which mitigates numerical oscillations in controlled settings, restoring exponential convergence. Our theoretical framework is validated through numerical experiments on a 2D flow past cylinder with Reynolds 1000, both at full and reduced order model levels.

**15:00 – Mattia Manucci (and B. Unger): Model order reduction for large-scale switched differential-algebraic equations**

We discuss a projection-based model order reduction (MOR) for large-scale systems of switched differential-algebraic equations (sDAEs), i.e.,

(1)

where is the switching signal, i.e., a piecewise constant function taking values in the index set , , and denote respectively, the *state*, the controlled *input*, and the measured *output*. We emphasize that the matrices for may be singular. Control systems of sDAEs may arise in modelling physical systems with algebraic constraints and piecewise time-dependent parameters, like Stokes control system with piecewise time-dependent diffusion. If (1) has to be evaluated repeatedly, one can rely on MOR and replace (1) by the *reduced-order model*

* *(2)

with , and for . Model reduction for systems of switched ordinary differential equations (sODEs) has been addressed in many works, see for instance [1,3,4] and reference therein, while, to the best of our knowledge, MOR of sDAEs only appears in [2] for a known switching signal and with several limitations to the large-scale setting, see [2, Sec. 5]. For our MOR scheme, we rely on a reformulation of (1) as a system of sODEs with state jumps at the switching times [2] and successively show that the method proposed in [3] can be successfully applied, in this generalized settings, to derive a reduced-order model for the set of generic admissible switching signals.

**References:**

[1] I. V. Gosea, M. Petreczky, A. C. Anntoulas, and C. Fiter. *Balanced truncation for linear switched systems. *Adv. Comput. Math., 44(6):1845–1886, 2018.

[2] M. S. Hossain and S. Trenn. *Model reduction for switched differential-algebraic equations with known switching signal.* Technical report, 2023.

[3] I. Pontes Duff, S. Grundel, and P. Benner. *New gramians for switched linear systems: Reachability, observability, and model reduction.* IEEE Trans. Automat. Control, 65(6):2526–2535, 2020.

[4] P. Schulze and B. Unger*. Model reduction for linear systems with low-rank switching.* SIAM J. Cont. Optim., 56(6):4365–4384, 2018.

**15:30 – Matteo Tomasetto (& Francesco Braghin, Andrea Manzoni): Real-time optimal control of parameterized systems by deep learning-based reduced order models**

Many optimal control problems require suitable strategies in order to steer instantaneously the considered dynamics. Moreover, the control action needs to be updated whenever the underlying scenario undergoes variations, as often happens in applications. Fullorder models based on, e.g., Finite Element Method, are not suitable for these settings due to the computational burden. In addition, conventional reduced order modeling techniques, such as the Reduced Basis method, are linear, intrusive, and usually not efficient in addressing nonlinear time-dependent dynamics. We thus propose a nonlinear, non-intrusive Deep Learning-based Reduced Order Modeling (DL-ROM) technique to control rapidly parametrized PDEs under different scenarios. After optimal snapshots generation, dimensionality reduction and neural networks training in the offline phase, optimal control strategies can be retrieved online in real-time for all the scenarios of interest. The speedup and the high accuracy of the proposed approach have been assessed on different PDE-constrained optimization problems, ranging from the minimization of energy dissipation under Navier-Stokes equations to thermal active cloaking.

**16:00 – Discussion session (Topic TBC)**