Abstract
A model of production funds acquisition, which includes two differential links of the zero order and two series-connected inertial links, is considered in a one-sector economy. Zero-order differential links correspond to the equations of the Ramsey model. These equations contain scalar bounded control, which determines the distribution of the available funds into two parts: investment and consumption. Two series-connected inertial links describe the dynamics of the changes in the volume of the actual production at the current production capacity. For the considered control system, the problem is posed to maximize the average consumption value over a given time interval. The properties of optimal control are analytically established using the Pontryagin maximum principle. The cases are highlighted when such control is a bang-bang, as well as the cases when, along with bang-bang (non-singular) portions, control can contain a singular arc. At the same time, concatenation of singular and non-singular portions is carried out using chattering. A bang-bang suboptimal control is presented, which is close to the optimal one according to the given quality criterion. A positional terminal control is proposed for the first approximation when a suboptimal control with a given deviation of the objective function from the optimal value is numerically found. The obtained results are confirmed by the corresponding numerical calculations.
Highlights
The elements of the zero order, first order, and second order are often found in nonlinear models of economic dynamics [1]
The link of the second order can be represented by an oscillatory link or by two connected inertial links
An example of a mathematical model containing inertial links is a model that takes into account the delay in the introduction of funds in the problem of optimizing the accumulation rate for the Ramsey model [1]
Summary
The elements of the zero order (multiplier, accelerator), first order (inertial links), and second order are often found in nonlinear models of economic dynamics [1]. The second equation refers to the oscillatory link characterizing the dynamics of the development of the introduced production capacities [1]. The equations of the dynamical change in the introduced production capacities of the Ramsey model [1,2,3,4,5] are used as dynamic links They contain a scalar bounded control u(t), which defines the distribution of the available funds into two parts: investment and consumption. The solution of the optimal control problem for this version of the model provides important information about the extreme value of the objective function and the type of optimal control This control can contain singular controls of the third order, the direct implementation of which in economic practice can be complicated by the presence of chattering regimes. The obtained properties of the optimal, suboptimal, and terminal controls for the considered model are confirmed by corresponding numerical calculations
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