Search

In this paper, we considered the problem of optimally controlling a twodimensional dynamical system until it reaches either of two boundaries. We consider a controlled dynamical system (X(t), Y(t)) which is a generalization of the classic twodimensional Kermack-McKendrick model for the spread of epidemics. Moreover, the system is subject to random jumps of fixed size according to a Poisson process. The system is controlled until the sum X(t) + Y(t) equal to either 0 or d (> 0) for the first time. Particular problems are solved explicitly.

Filtered Poisson processes are used as models in various applications, in particular in statistical hydrology. In this paper, controlled filtered Poisson processes are considered. The aim is to minimize the expected time that the process will spend in the continuation region. The dynamic programming equation satisfied by the value function is derived. To obtain the value function, and hence the optimal control, a non-linear integro-differential equation must be solved, subject to the appropriate boundary conditions. Various cases for the size of the jumps are treated and explicit results are obtained. [ABSTRACT FROM AUTHOR]

The state of a machine is modeled as a controlled continuous-time Markov chain. Moreover, the machine is being serviced at random times. The aim is to maximize the time until the machine must be repaired, while taking the maintenance costs into account. The dynamic programming equation satisfied by the value function is derived, enabling optimal decision-making regarding inspection rates, and special problems are solved explicitly. This approach minimizes direct maintenance costs along with potential failure expenses, establishing a robust methodology for determining inspection frequencies in reliability-centered maintenance. The results contribute to the advancement of maintenance strategies and provide explicit solutions for particular cases, offering ideas for application in reliability engineering. [ABSTRACT FROM AUTHOR]

Suppose that in an M / G / 1 waiting queue, the server can choose between two service time distributions. We look for the choice that enables us to minimize the expected value of a cost criterion that takes into account the cost incurred by working faster and the time needed to empty the waiting line. The random final time is the first time there is no customer waiting for service. When the service times are exponential random variables, we can appeal to dynamic programming to obtain the optimal solution. In the general case, conditional probability is used. Particular problems in which the capacity of the system is finite are solved explicitly. [ABSTRACT FROM AUTHOR]

The problem of controlling a compound Poisson process until it leaves an interval is considered. This type of problem is known as a homing problem. To determine the value of the optimal control, we must solve a nonlinear integro-differential equation. Exact and explicit solutions are obtained for two possible jumps size distributions.

Let $ (X_n,Y_n) $ (X n , Y n) be a two-dimensional discrete-time controlled Markov chain. The problem of optimally controlling the chain until the sum $ X_n+Y_n $ X n + Y n is equal to either of two constants is considered. The aim is to leave the continuation region as soon as possible. The optimal control is found in two particular cases by solving the partial difference equation satisfied by the value function. [ABSTRACT FROM AUTHOR]

Let ${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$ , where Y (t) is a one-dimensional diffusion process, and let $\tau(x,y)$ be the first time the process (X (t), Y (t)), starting from (x , y), leaves a subset of the first quadrant. The problem of computing the probability $p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p (x , y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of $\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of $\tau(x,y)$ can also be derived. [ABSTRACT FROM AUTHOR]

The streamflow of a river is modelled as a jump-diffusion process. The jump size is distributed as an exponential random variable. The various parameters of the model are estimated by using the method of moments. An application to the Mississippi River is presented. [ABSTRACT FROM AUTHOR]

The problem of optimally controlling a standard Brownian motion until a fixed final time is considered in the case when the final cost function is an even function. Two particular problems are solved explicitly. Moreover, the best constant control as well as the best linear control are also obtained in these two particular cases.

A boundary value problem for a non-linear difference equation of order three is considered. We show that this equation can be interpreted as the equation satisfied by the value function in a stochastic optimal control problem. We thus obtain an expression for the solution of the non-linear difference equation that can be used to find an explicit solution to this equation. An example is presented. [ABSTRACT FROM AUTHOR]

Various linear and non-linear first-order differential equations with different arguments are considered. Exact solutions to these equations are provided. Systems of two coupled linear first-order differential equations are also solved explicitly and exactly. [ABSTRACT FROM AUTHOR]

An analytical solution is found to the problem of maximising the time spent in the first quadrant by the two-dimensional diffusion process (X(t), Y(t)), where Y(t) is a controlled Brownian motion and X(t) is proportional to its integral. Moreover, we force the process to exit the first quadrant through the y-axis. This type of problem is known as LQG homing and is very difficult to solve explicitly, especially in two or more dimensions. Here the partial differential equation satisfied by a transformation of the value function is solved by making use of the method of separation of variables. The exact solution is expressed as an infinite sum of Airy functions.