Optimization with affine homogeneous quadratic integral inequality constraints

G. Fantuzzi, A. Wynn, P. J. Goulart and A. Papachristodoulou

IEEE Transactions on Automatic Control, vol. 62, no. 12, pp. 6221-6236, December 2017.
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  author = {G. Fantuzzi and A. Wynn and P. J. Goulart and A. Papachristodoulou},
  title = {Optimization with affine homogeneous quadratic integral inequality constraints},
  journal = {IEEE Transactions on Automatic Control},
  year = {2017},
  volume = {62},
  number = {12},
  pages = {6221-6236},
  url = {http://dx.doi.org/10.1109/TAC.2017.2703927},
  doi = {10.1109/TAC.2017.2703927}

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 given homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Problems of this type often arise when studying dynamical systems governed by partial differential equations. First, we derive a hierarchy of outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that a convergent, non-decreasing sequence of lower bounds for the optimal cost value can be computed by solving a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, which enable us to formulate SDPs to compute upper bounds for the optimal cost, as well as to compute a strictly feasible point for the integral inequality. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to the MATLAB optimization toolbox YALMIP. We demonstrate our techniques by solving typical problems that arise in the context of stability analysis for dynamical systems governed by PDEs.