*Maximum Principle for Innite Horizon Discrete Time Optimal Control Problems*

**Mária Holecyová**

PhD thesis advisor: **Pavel Brunovský **

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**Summary**

The aim of this thesis is a method of deriving necessary conditions of the Potryagin maximum principle type for innite-horizon discrete-time optimal control problems with discount. Due to the discounted objective function, control and state variables are considered to be bounded sequences. We employ the tools of functional analysis and properties of linear difference systems.

Firstly, we prove Fréchet differentiability of the objective function which allows us to carry out a standard method of obtaining necessary conditions of optimality of variational type. Then we apply the closed range theorem and formulate maximum principle in functional form with adjoint variable from the space. Then we show that it can be rewritten to the standard form of Potryagin maximum principle for adjoint variable.

The most significant results are conditions under which the assumptions of the closed range theorem are satisfied. For a problem with linear dynamics we require that the matrix A has no eigenvalues on the unit circle and in case of general dynamics we formulate exponential dichotomy as an assumption. We present special cases in which exponential dichotomy can be effectively verified. In addition, on a simple example we show that without exponential dichotomy the assumption of closed range probably may not hold.

**Keywords:** optimal control, discrete time, infinite horizon, Pontryagin maximum