A Class of Linear Quadratic Dynamic Optimization Problems with State Dependent Constraints

In this paper, we analyse a wide class of discrete time one-dimensional dynamic optimization problems - with strictly concave current payoffs and linear state dependent constraints on the control parameter as well as non-negativity constraint on the state variable and control. This model suits well economic problems like extraction of a renewable resource (e.g. a fishery or forest harvesting). The class of sub-problems considered encompasses a linear quadratic optimal control problem as well as models with maximal carrying capacity of the environment (saturation). This problem is also interesting from theoretical point of view - although it seems simple in its linear quadratic form, calculation of the optimal control is nontrivial because of constraints and the solutions has a complicated form. We consider both the infinite time horizon problem and its finite horizon truncations.