arXiv subjects

Milan Korda

Publications and source records attributed to Milan Korda.

At least 19 recordsLinked to original sources

arXiv2026

Concentration of measure-valued solutions for semilinear parabolic equations

The moment-sum-of-squares hierarchy provides a powerful framework for solving non-convex optimal control problems by constructing a sequence of convex semidefinite relaxations. However, when extending these methods to nonlinear partial differential equations (PDEs), a fundamental challenge is the potential existence of a relaxation gap, where the solution to the linear measure formulation using occupation measures fails to correspond to a classical physical solution of the original PDE. In this paper, we prove the absence of a relaxation gap for scalar semilinear parabolic PDEs of the reaction-diffusion type. We do so by showing that each solution to the linear measure equation gives rise to an energy measure-valued (emv) solution in the space of Young measures satisfying suitable energy identities. We then prove that any such emv solution concentrates on the solution to the nonlinear PDE, provided the latter exists and is unique. To the best of our knowledge, this is the first concentration result of this kind for measure-valued solutions of reaction-diffusion PDEs.

math.OC↗

arXiv2026

Composition and tensor train structure in polynomial optimization

We study polynomial optimization problems whose objective has a composition or tensor train structure. These polynomials can be evaluated as a sequence of maps, giving rise to intermediate variables (``states'') of dimension lower than the ambient dimension. Structures like these arise naturally in dynamical systems, Markov chains, and neural networks. We develop two moment-SOS (sums of squares) hierarchies that exploit this composition structure in different ways. The first one, termed state-lifting chordal, is based on the correlative sparsity graph of the problem. The second one, termed state-lifting push-forward, encodes the structure at the level of the measures directly. Numerical experiments demonstrate that the proposed methods can compute certified bounds for problems with hundreds or even a thousand variables. To illustrate the versatility of the hierarchies we apply them to Markov chain optimization, quantum optimal control, and neural networks.

math.OC↗

arXiv2026

Semidefinite relaxations for nonlinear elasticity with energies convex in the Cauchy-Green strain tensor

In nonlinear elasticity, finding the deformation of a material which minimizes a given stored energy density is a challenging calculus of variations problem which may fail to have minimizers: the energy optimal material forms infinitely fine microstructures (wrinkles) rather than deforming smoothly. In the case where the energy function is non-convex but frame indifferent and convex with respect to the Cauchy-Green strain tensor, we use the standard Le Dret-Raoult semidefinite projection formula for the quasiconvex envelope of the energy function to prove that there is no relaxation gap between the original non-convex calculus of variations problem and its linear moment formulation based on occupation measures. This implies convergence of the Lasserre moment-sum-of-squares (SOS) hierarchy and provides a computationally efficient, mesh-free numerical method that, unlike the finite element method, avoids undesirable mesh-dependent artifacts. Under the additional condition that the boundary condition is linear and the function is SOS convex in the strain tensor, we show that the first relaxation of the Lasserre hierarchy is exact. In other words, computing the quasiconvex envelope at a point boils down to solving a small convex semidefinite optimization problem.

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arXiv2025

Global optimization of low-rank polynomials

This work considers polynomial optimization problems where the objective admits a low-rank canonical polyadic tensor decomposition. We introduce LRPOP (low-rank polynomial optimization), a new hierarchy of semidefinite programming relaxations for which the size of the semidefinite blocks is determined by the canonical polyadic rank rather than the number of variables. As a result, LRPOP can solve low-rank polynomial optimization problems that are far beyond the reach of existing sparse hierarchies. In particular, we solve problems with up to thousands of variables with total degree in the thousands. Numerical conditioning for problems of this size is improved by using the Bernstein basis. The LRPOP hierarchy converges from below to the global minimum of the polynomial under standard assumptions.

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arXiv2025

Set-Valued Koopman Theory for Control Systems

In this paper, we introduce a new notion of Koopman operator which faithfully encodes the dynamics of controlled systems by leveraging the tools of set-valued analysis. In this context, we propose generalisations of the Liouville and Perron-Frobenius operators, and show that they respectively coincide with proper set-valued analogues of the infinitesimal generator and dual operator of the Koopman semigroup. We also give meaning to the spectra of these set-valued maps and prove an adapted version of the classical spectral mapping theorem relating the eigenvalues of a semigroup with those of its generator. Our approach provides theoretical justifications for existing practical methods in the Koopman community that study control systems by bundling together the Koopman and Liouville operators associated with different control inputs.

math.OC↗

arXiv2025

Data-driven control of a magnetohydrodynamic flow

We demonstrate the feedback control of a weakly conducting magnetohydrodynamic (MHD) flow via Lorentz forces generated by externally applied electric and magnetic fields. Specifically, we steer the flow of an electrolyte toward prescribed velocity or vorticity patterns using arrays of electrodes and electromagnets positioned around and beneath a fluid reservoir, with feedback provided by planar particle image velocimetry (PIV). Control is implemented using a model predictive control (MPC) framework, in which control signals are computed by minimizing a cost function over the predicted evolution of the flow. The predictor is constructed entirely from data using Koopman operator theory, which enables a linear representation of the underlying nonlinear fluid dynamics. This linearity allows the MPC problem to be solved by alternating between two small and efficiently solvable convex quadratic programs (QPs): one for the electrodes and one for the electromagnets. The resulting controller runs in a closed loop on a standard laptop, enabling real-time control of the flow. We demonstrate the functionality of the approach through experiments in which the flow is shaped to match a range of reference velocity fields and a time-varying vorticity field.

physics.flu-dyn↗

arXiv2025

Polynomial argmin for recovery and approximation of multivariate discontinuous functions

We propose to approximate a (possibly discontinuous) multivariate function f (x) on a compact set by the partial minimizer arg miny p(x, y) of an appropriate polynomial p whose construction can be cast in a univariate sum of squares (SOS) framework, resulting in a highly structured convex semidefinite program. In a number of non-trivial cases (e.g. when f is a piecewise polynomial) we prove that the approximation is exact with a low-degree polynomial p. Our approach has three distinguishing features: (i) It is mesh-free and does not require the knowledge of the discontinuity locations. (ii) It is model-free in the sense that we only assume that the function to be approximated is available through samples (point evaluations). (iii) The size of the semidefinite program is independent of the ambient dimension and depends linearly on the number of samples. We also analyze the sample complexity of the approach, proving a generalization error bound in a probabilistic setting. This allows for a comparison with machine learning approaches.

math.NA↗

arXiv2025

Solving unbounded optimal control problems with the moment-SOS hierarchy *

The behaviour of the moment-sums-of-squares (moment-SOS) hierarchy for polynomial optimal control problems on compact sets has been explored to a large extent. Our contribution focuses on the case of non-compact control sets. We describe a new approach to optimal control problems with unbounded controls, using compactification by partial homogenization, leading to an equivalent infinite dimensional linear program with compactly supported measures. Our results are closely related to the results of a previous approach using DiPerna-Majda measures. However, our work provides a sound proof of the absence of relaxation gap, which was conjectured in the previous work, and thereby enables the design of a moment-sum-of-squares relaxation with guaranteed convergence.

math.OC↗

arXiv2025

Optimal Control of 1D Semilinear Heat Equations with Moment-SOS Relaxations

We use moment-SOS (Sum Of Squares) relaxations to address the optimal control problem of the 1D heat equation perturbed with a nonlinear term. We extend the current framework of moment-based optimal control of PDEs to consider a quadratic cost on the control. We develop a new method to extract a nonlinear controller from approximate moments of the solution. The control law acts on the boundary of the domain and depends on the solution over the whole domain. Our method is validated numerically and compared to a linear-quadratic controller.

math.OC↗

arXiv2025

Young measure relaxation gaps for controllable systems with smooth state constraints

In this article, we tackle the problem of the existence of a gap corresponding to Young measure relaxations for state-constrained optimal control problems. We provide a counterexample proving that a gap may occur in a very regular setting, namely for a smooth controllable system state-constrained to the closed unit ball, provided that the Lagrangian density (i.e., the running cost) is non-convex in the control variables. The example is constructed in the setting of sub-Riemannian geometry with the core ingredient being an unusual admissible curve that exhibits a certain form of resistance to state-constrained approximation. Specifically, this curve cannot be approximated by neighboring admissible curves while obeying the state constraint due to the intricate nature of the dynamics near the boundary of the constraint set. This example therefore demonstrates the impossibility of Filippov-Wazewski type approximation in the presence of state constraints. Our example also presents an occupation measure relaxation gap.

math.OC↗

arXiv2025

Sufficient conditions for the absence of relaxation gaps in state-constrained optimal control

This work presents new sufficient conditions for the absence of a gap corresponding to Young measure and occupation measure relaxations for constrained optimal control problems. Unlike existing conditions, these sufficient conditions do not rely on convexity of the Lagrangian or the set of admissible velocities. We use these conditions to derive new bounds for the size of the relaxation gap.

math.OC↗

arXiv2025

Clarifying the effect of mean subtraction on Dynamic Mode Decomposition

Any autonomous nonlinear dynamical system can be viewed as a superposition of infinitely many linear processes, through the so-called Koopman mode decomposition. Its data-driven approximation- Dynamic Mode Decomposition (DMD)- has been extensively developed and deployed across a plethora of fields. In this work, we study the effect of subtracting the temporal mean on the DMD approximation, for observables possessing only a finite number of Koopman modes. Pre-processing time-sequential training data by removing the temporal mean has been a point of contention in the Companion matrix formulation of DMD. This stems from the potential of said pre-processing to render DMD equivalent to a temporal Discrete Fourier Transform (DFT). We prove that this equivalence is impossible when the training data is linearly consistent and the order of the DMD model exceeds the number of Koopman modes. Since model order and training set size are synonymous in this variant of DMD, the parity of DMD and DFT can, therefore, be indicative of inadequate training data.

math.DS↗

arXiv2024

Learning Parametric Koopman Decompositions for Prediction and Control

We present an approach to construct approximate Koopman-type decompositions for dynamical systems depending on static or time-varying parameters. Our method simultaneously constructs an invariant subspace and a parametric family of projected Koopman operators acting on this subspace. We parametrize both the projected Koopman operator family and the dictionary that spans the invariant subspace by neural networks and jointly train them with trajectory data. We show theoretically the validity of our approach, and demonstrate via numerical experiments that it exhibits significant improvements over existing methods in solving prediction problems, especially those with large state or parameter dimensions, and those possessing strongly non-linear dynamics. Moreover, our method enables data-driven solution of optimal control problems involving non-linear dynamics, with interesting implications on controllability.

math.OC↗

arXiv2024

Towards Optimal Spatio-Temporal Decomposition of Control-Related Sum-of-Squares Programs

This paper presents a method for calculating the Region of Attraction (ROA) of nonlinear dynamical systems, both with and without control. The ROA is determined by solving a hierarchy of semidefinite programs (SDPs) defined on a splitting of the time and state space. Previous works demonstrated that this splitting could significantly enhance approximation accuracy, although the improvement was highly dependent on the ad-hoc selection of split locations. In this work, we eliminate the need for this ad-hoc selection by introducing an optimization-based method that performs the splits through conic differentiation of the underlying semidefinite programming problem. We provide the differentiability conditions for the split ROA problem, prove the absence of a duality gap, and demonstrate the effectiveness of our method through numerical examples.

math.OC↗

arXiv2024

Sparse decompositions of nonlinear dynamical systems and applications to moment-sum-of-squares relaxations

In this paper, we propose a general sparse decomposition of dynamical systems provided that the vector field and constraint set possess certain sparse structures, which we call subsystems. This notion is based on causal dependence in the dynamics between the different states. This results in sparse descriptions for fundamental problems from nonlinear dynamical systems: region of attraction, maximum positively invariant set, and global attractor. The decompositions can be paired with any method for computing (outer) approximations of these sets to reduce the computation to lower dimensional systems. This is illustrated by methods from previous work based on infinite-dimensional linear programming. This exhibits one example where the curse of dimensionality is present and hence dimension reduction is crucial. In this context, for polynomial dynamics, we show that these problems admit a sparse sum-of-squares (SOS) approximation with guaranteed convergence such that the number of variables in the largest SOS multiplier is given by the dimension of the largest subsystem appearing in the decomposition. The dimension of such subsystems depends on the sparse structure of the vector field and the constraint set; if the dimension of the largest subsystem is small compared to the ambient dimension, this allows for a significant reduction in the computation time of the SOS approximations. Numerical examples accompany the approach.

math.OC↗

arXiv2023

Invariant Consistent Dynamic Mode Decomposition

Any deterministic autonomous dynamical system may be globally linearized by its' Koopman operator. This object is typically infinite-dimensional and can be approximated by the so-called Dynamic Mode Decomposition (DMD). In DMD, the central idea is to preserve a fundamental property of the Koopman operator: linearity. This work augments DMD by preserving additional properties like functional relationships between observables and consistency along geometric invariants. The first set of constraints provides a framework for understanding DMD variants like Higher-order DMD and Affine DMD. The latter set guarantees the estimation of Koopman eigen-functions with eigen-value 1, whose level sets are known to delineate invariant sets. These benefits are realized with only a minimal increase in computational cost, primarily due to the linearity of constraints.

math.DS↗

arXiv2023

Exploiting Term Sparsity in Moment-SOS hierarchy for Dynamical Systems

In this paper, we develop a dynamical system counterpart to the term sparsity sum-of-squares (TSSOS) algorithm proposed for static polynomial optimization. This allows for computational savings and improved scalability while preserving convergence guarantees when sum-of-squares methods are applied to problems from dynamical systems, including the problems of approximating region of attraction, the maximum positively invariant set, and the global attractor. At its core, the method exploits the algebraic structure of the data, thereby complementing existing methods that exploit causality relations among the states of the dynamical system. The procedure encompasses sign symmetries of the dynamical system as was already revealed for polynomial optimization. Numerical examples demonstrate the efficiency of the approach in the presence of this type of sparsity.

math.OC↗

arXiv2023

Converging outer approximations to global attractors using semidefinite programming

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

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