We establish global exponential turnpike properties for quadratic optimal tracking problems governed by the one-dimensional viscous Burgers equation with localized internal control. For every initial datum, finite-horizon optimal solutions approach the unique optimal periodic regime when the periodic tracking target is sufficiently small; the zero-target case yields a global steady turnpike at the origin, with no smallness assumption on the initial datum. To our knowledge, these are the first global exponential turnpike results for the viscous Burgers equation. The proof combines a local exponential turnpike, obtained through strict convexity and periodic Riccati theory, with a parabolic dissipation argument that provides an absorbing time independent of the horizon.
Open access
2 source records
Stability and Controllability of Differential Equations
The Debreu Koopmans theorem restricts separable aggregation to at most one nonconvex component. We solve this by proving that a separable, additive or multiplicative, function is star quasiconvex, those with star shaped sublevel sets about minimizers, if and only if each component is star quasiconvex. This immediately yields star quasiconvexity of separable sums of quasiconvex functions, formally bridging diversification theory with the S shaped value functions of Prospect Theory. Furthermore, we develop a complete calculus, monotonic composition, pointwise minima, quasi arithmetic means, and we apply it to Cobb-Douglas functions, multifactor risk models, and constant function market makers in decentralized finance. Star quasiconvexity thus provides a unified framework for applications in optimization and economic modeling beyond the classical Debreu Koopmans constraint. The introduction discuss economic motivations.
The Debreu–Koopmans theorem [14] restricts separable aggregation to at most one nonconvex component. We solve this by proving that a separable (additive or multiplicative) function is star quasiconvex (those with star-shaped sublevel sets about minimizers) if and only if each component is star quasiconvex. This immediately yields star quasiconvexity of separable sums of quasiconvex functions, formally bridging diversification theory with the S-shaped value functions of Prospect Theory. Furthermore, we develop a complete calculus (monotonic composition, pointwise minima, quasi-arithmetic means) and we apply it to Cobb-Douglas functions, multi-factor risk models, and constant function market makers in decentralized finance. Star quasiconvexity thus provides a unified framework for economic modeling beyond the classical Debreu–Koopmans constraint.
This paper presents a new theory, known as robust dynamic programming, for a class of continuous-time dynamical systems. Different from traditional dynamic programming (DP) methods, this new theory serves as a fundamental tool to analyze the robustness of DP algorithms, and, in particular, to develop novel adaptive optimal control and reinforcement learning methods. In order to demonstrate the potential of this new framework, two illustrative applications in the fields of stochastic and decentralized optimal control are presented. Two numerical examples arising from both finance and engineering industries are also given, along with several possible extensions of the proposed framework.
We study a two-level system having N local systems in the lower level subordinate to a central system in the higher one, such that both central and local systems have decision-making units. The central system is a coordinating agency and the local ones are semi-autonomous operating devisions. The basic principle of planning for this organization is that the central system allocates resources so as to optimize its own objective, while the local ones optimize their own objectives using the given resources. A local objective function, fn, is a function of the lower level decision variable vector x=(x1,・・・, xN) and the higher level one a=(a1,・・・, aN), where an is a resource vector allocated to the local system n. Since the functions ■ are mutually independent, the lower level composes a multi-objective system, in which the lower level decision-makers minimize a vector objective function f =(f1,・・・,fN) with respect to x in cooperation with each other. Thus, the lower level generates a set of noninferior (i.e. Pareto optimal) solutions ■(a) being parametric with respect to a. The central decision-maker, then, chooses the optimal resource allocation a⁰ and the best noninferior solution ■⁰ corresponding to a⁰ from among a set of ■(a). The above problem becomes a decentralized two-level optimization, when the local system contains only its own variables (xn, an). Several theorems and iterative algorithms for the formulated problems are obtained by use of mathematical programming techniques.