Abstract
We introduce a new technique to optimize a linear cost function subject to a one-dimensional affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in stability analysis, input-to-state/output analysis, and control of many systems governed by partial differential equations (PDEs), in particular fluid dynamical systems. First, we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that under mild assumptions a convergent, non-decreasing sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can also be computed with SDPs. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our techniques by solving problems arising from the stability analysis of PDEs.
Highlights
Analysis and control of systems governed by partial differential equations (PDEs) are fundamental problems in physics and engineering, but are challenging because the system state is a function w of both the time t and the spatial position vector x, and as such it belongs to an infinite-dimensional space.In an effort to reduce the conservativeness introduced by finite-dimensional approximations, recent years have seen the development of analytical techniques that consider directly the infinite-dimensional PDEs, and lead to consideration of integral inequalities
We present QUINOPT, an add-on to YALMIP [25, 26] to aid the formulation of the semidefinite programs (SDPs) relaxations outlined above, and use it to solve examples that demonstrate the advantages of our approach compared to the SOS method of [37]
Our results demonstrate that our SDP relaxations accurately approximate (53); this is significant for the inner SDPs, which rely on typically conservative estimates and for which we cannot prove convergence
Summary
Analysis and control of systems governed by partial differential equations (PDEs) are fundamental problems in physics and engineering, but are challenging because the system state is a (vector-valued) function w of both the time t and the spatial position vector x, and as such it belongs to an infinite-dimensional space (e.g. a Sobolev space). The computational cost of designing optimal control policies for systems with complex dynamics, such as turbulent flows, may be reduced by requiring the control law to minimize an upper bound on the objective function rather than the objective itself [20, 21, 23, 24], and in the case of PDEs such upper bounds can be found by solving suitable integral inequalities [8, 9, 11,12,13, 19]. Could be solved (in principle) by first computing the minimizer w⋆ of Fγ as a function of γ using the calculus of variations [10, 18], and minimizing the augmented Lagrangian L(γ) = cT γ − λFγ{w⋆}, where the Lagrange multiplier λ ≥ 0 is chosen to enforce the integral inequality This strategy has been successfully applied to some problems in fluid dynamics
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