^{1}

^{2}

^{1}

^{2}

For the particularity of warfare hybrid dynamic process, a class of warfare
hybrid dynamic systems is established based on Lanchester equation in a

In 1914, Lanchester [

To establish the mathematical model describing warfare process is the basis for researching quantitatively decision-making problems in conflicts. So far, warfare system models based on Lanchester equation have penetrated into many fields of military problems; the research core mainly focuses on extending and modifying Lanchester equation. For example, in [

In fact, warfare dynamic process is a hybrid dynamic process, which is characterized by the interaction of continuous time dynamic process (described, e.g., fighting strength changes) and discrete event dynamic process (described, e.g., fighting strength scheduling, variable tactics). Thus, how to establish warfare hybrid dynamic system model, which integrates the discrete events with continuous time, has become a problem to be solved urgently. In [

Optimal control problem of warfare dynamic system has been an area of considerable research interest and has been an absolutely necessary tache on using Lanchester equation to research the tactic decision-making problem. So far, a wide variety of research achievements on this problem have been obtained, such as. In [

The paper is organized as follows. In Section

In order to directly understand the basic frame of warfare hybrid dynamic system, the evolution analysis of warfare process is given in Figure

Evolution analysis of warfare hybrid dynamic process in a

Firstly, let

firstly,

at the time

at the time

the above-mentioned process continues.

There is no difficulty in deducing the conclusion that variable tactics of the decision-maker

The warfare process evolution also involves the continuous control of force strengths on both combat units. Based on decision-maker’s instructions and detected situation, each combat unit adjusts the control variables with the purpose of changing the force strengths; however, the warfare task is certain herein. It falls into a category of continuous control process with the systematic structure unchanged. Therefore, warfare dynamic process can be considered as warfare hybrid dynamic system, which is that force strengths change on both sides from one continuous system to another via certain variable tactics (those that change combat encounter), and every variable tactic happens; the whole system operates following the later’s rules.

Inspired by [

Suppose that variable tactics happen at time

Motivated by the above discussions, a class of warfare hybrid dynamic systems based on Lanchester equation can be established as follows:

From Assumption

The values of switch variable

From Remarks

In this section, the optimal variable tactics control problem of warfare hybrid dynamic system in a

The values of switch variables

From Assumption

The objective function associated with system (

Now, the optimal variable tactics control problem can be described as follows. The attacking side

In this subsection, we analyze the conditions of the optimal variable tactics and give a quantitative analysis of the variable tactics process. Finally, a solving method for the optimal control strategies is designed.

For the above optimal control problem, we introduce the adjoint function as follows:

From (

From (

Since

Based on the above analysis, we discuss the variable tactics process about the attacking side

If there exist at least two functions

From (

From (

Now, we will investigate that

According to Theorem

Using

Using

The aforementioned process continues. When the conditions of tactic change cannot hold in the time interval

Sorting

As an example of the consequences of the optimal control problem, we take the

The objective function associated with the system (

Solving the optimal control problem by Matlab Toolbox yields that when

Figure

Strength change curves of each combat unit on both sides.

In this paper, we established a class of warfare hybrid dynamic systems based on Lanchester equation in a battle between an attacker with one type of force and a defender with

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to express their deepest gratitude to Dr. Qiu Jianlong, Dr. Jianwei Zhou, and Dr. Chen Xiao for their helpful suggestions on the English writing of the revised version. This work was supported in part by the Applied Mathematics Enhancement Program (AMEP) of Linyi University and the National Natural Science Foundation of China, under Grant nos. 61273012, 11301252, 61304023, and 11201212, and by a Project of Shandong Province Higher Educational Science and Technology Program under Grant nos. J13LI11 and J12LI58.