Math

Browse all posts in the Math category.

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11JUN 2026
Math
Sums-of-squares Polyconvexity
Sums-of-squares Polyconvexity Find this post at FAU DCN-AvH Math & Research posts. 1 Introduction Polyconvex functions are functions of a matrix variable characterized by a convex dependence on the minors of the matrix. This generalization of convexity is fundamental in the Calculus of Variations, where it provides a sufficient condition for the well-posedness of […]
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31MAY 2026
Math
Activation Saturation and Jacobian Attenuation in Neural Ordinary Differential Equations
Neural Ordinary Differential Equations (Neural ODEs) offer a principled way to model continuous dynamical systems with neural networks [1]. The hidden state evolves according to where is a multilayer perceptron with trained weights . Once training is finished, is fixed and will not change. The network no longer adapts—it defines a static vector field on […]
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01APR 2026
Math
Optimal control for renormalized solutions of nonlinear evolution equations
1 Introduction We aim to develop a comprehensive theory of optimal control for nonlinear parabolic equations of Leray-Lions type, whose solutions may fail to exist in the classical weak sense for low regularity data. In this case, the appropriate notion of solution is the renormalized solution, introduced by Lions and Di Perna [8] for […]
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17MAR 2026
Math
Network design and control: Shape and topology optimization for the turnpike property for the wave equation
Network design and control: Shape and topology optimization for the turnpike property for the wave equation 1 Introduction We consider two optimal control problems. The first problem, denoted by , is an optimal control problem governed by an evolution equation. The second problem, denoted by , is the optimal control problem for the associated […]
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16MAR 2026
Math
Reinforcement Learning and LQR with special control structure: switched and multilevel systems
Reinforcement Learning and LQR with special control structure: switched and multilevel systems 1 Introduction In this post we study the well-known linear-quadratic regulator problem in continuous time where , , , , and is the initial state. The goal is to choose a control in such a way that at time , we have […]
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11MAR 2026
Math
Exact Controllability of Stochastic First-Order Multi-Dimensional Hyperbolic Systems
Exact Controllability of Stochastic First-Order Multi-Dimensional Hyperbolic Systems In the real world, the evolution of many physical quantities can be described by first-order hyperbolic systems. Notable examples include the Saint-Venant equations for open channels, the Aw-Rascle model for road traffic, gas dynamics, and shallow water equations. Over the past few decades, the boundary control theory […]Exact Controllability of Stochastic First-Order Multi-Dimensional Hyperbolic Systems In the real world, the evolution of many physical quantities can be described by first-order hyperbolic systems. Notable examples include the Saint-Venant equations for open channels, the Aw-Rascle model for road traffic, gas dynamics, and shallow water equations. Over the past few decades, the boundary control theory […]
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