In this work we propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in environmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save computational time, we rely on reduced basis techniques as a suitable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems and the saddle-point structure of their optimality system. Then, we propose a POD–Galerkin reduction of the optimality system. We test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy, and a solution tracking governed by quasi-geostrophic equations describing the North Atlantic Ocean dynamics. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale, as well as physical meaning. The quasi-geostrophic optimal control problem is also presented in its nonlinear version.
- Environmental marine applications
- PDE state equations
- Parametrized optimal control problems
- Proper orthogonal decomposition
- Quasi-geostrophic equation
- Reduced order methods