Optimization with affine homogeneous quadratic integral inequality constraints

Journal article

We introduce a new technique to optimize a linear cost function subject to an 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 control and stability or input-to-state/output analysis of systems governed by partial differential equations (PDEs), particularly fluid dynamical systems. We derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that a convergent sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). We also 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 be computed with SDPs. We present QUINOPT, an open-source add-on to YALMIP to aid the formulation and solution of our SDPs, and demonstrate our techniques on problems arising from the stability analysis of PDEs.

Authors: Fantuzzi, G; Wynn, A; Goulart, P; Papachristodoulou, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Electrical and Electronics Engineers
• Journal: IEEE Transactions on Automatic Control
• Volume: 62
• Issue: 12
• Pages: 6221-6236
• Publication date: 2017-06-26
• DOI: 10.1109/TAC.2017.2703927
• ISSN: 0018-9286
• Keywords: aerospace electronics; symmetric matrices; sum-of-squares optimization; optimization; partial differential equations; semidefinite programming; integral equations; partial differential equations (PDEs); integral inequalities; stability analysis; upper bound