Optimal control problems for evolution systems belong to a very active area of research due to a large number of applications and interesting mathematical properties. In the case of infinite-dimensional setting, when the dynamics of the system are described by an operator acting on an infinite-dimensional space, numerical solving an optimal control problem involves the procedure of discretization, a procedure to approximate the solution of the problem to a finite-dimensional space. There are two approaches to finding the approximate solution of the problem: First discretise, then optimise and First optimise, then discretise. The former approach to construct an approximate solution is to substitute all appearing function spaces by finite-dimensional spaces and all appearing operators by operators on finite-dimensional spaces, i.e. matrices, and then solve the corresponding finite-dimensional problem. In our work we plan to follow the latter approach, which involves the application of optimization techniques on the original, infinite-dimensional problem, in order to find optimality conditions for the problem, and then to discretize these conditions. What will distinguish our approach from the other state-of-the-art techniques (e.g. primal-dual active strategy, variational discretisation concept and various gradient techniques) is the use of the functional calculus together with standard techniques from convex analysis. This will allow us to construct a closed form expression of the solution of the problem in the form of a linear system Ax = b, where both A and b are functions of the state dynamics operator. This expression for the solution will allow on the one hand to efficiently analyse various properties of the solution like sensitivity and, on the other hand, it will enable the use of state-of-the-art numerical techniques for the approximation of functions of the operators, like RKFIT. In this way, we will be able to choose the discretisation procedure which will be tailored to the specific problem at hand, not only taking into account the properties of the state dynamics operator, but also the other parameters of the optimal control problem as well. To the best of our knowledge, the other contemporary techniques are not suitable for such a granular approach to the discretisation. Such an approach proved successful in solving the optimal control problem in initial conditions, see L. Grubišić, M. Lazar, I. Nakić, M. Tautenhahn: Optimal control of parabolic equations – a spectral calculus based approach.

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