Abstract

This work introduce a difference controls of players by using a new control method to completing a pursuit game. We study pursuit game problems for controlled partial differential equations of the parabolic type. We proved a theorem on pursuit game with mixed constraints, where pursuers control are subjected to integral (geometric) constraint and geometric (integral) constraint are imposed on evaders control. Moreover, we established the sufficient conditions for which pursuit is possible in the game considered.

Subject Classification: 49N70, 49N75, 91A23

1. Introduction

Let A be a differential elliptic operator in the space of the form

(1)

The domain of definition of the operator is (space of twice continuously differentiable functions), is a bounded domain with piece-wise smooth boundary in and is a measurable bounded function satisfying the conditions and

Suppose that,

is the lateral surface of the open cylinder in , where the boundary of the domain is assumed to be piecewise smooth.

Further, recall ¹ that is the Hilbert space of elements ofwhose first-order generalized derivatives are square integrable on is the subspace of in which smooth compactly supported functions form a dense subset, is the Hilbert space of elements of whose generalized derivatives are square integrable on, andis the subspace of in which smooth functions vanishing near form a dense set.

We consider a controlled system described by the parabolic equation:

(2)

It was proved in ¹ that if the above-mentioned conditions are satisfied, then problem (2) has a unique solutionin the class for arbitrary are the control functions (parameters) of the first player (the pursuer) and the second player (the evader), respectively. The solution has the form

(3)

where the functionsform a solution of the following infinite system of differential equations and initial conditions:

(4)

where are the generalized eigenvalues of the operator A [see ⁴], (all these eigenvalues are negative, and as ), the functions form an orthonormal complete system of generalized eigenfunctions of A in . and are the Fourier coefficients of and respectively, in the system .

2. Preliminaries

We assume that the pursuit game problem with mixed constraints on controls of players. The controls functions and subject to the following systems of inequalities.

(5)

(6)

where, and are nonnegative constants.

Definition 2.1. In the pursuit game (2), (5), [respectively, (2), (6)], it is possible to complete the pursuit from an initial position if there exists and a function such that the following assertions are valid for an arbitrary function satisfying the condition

(i)

(ii) the solution of problem (2), where and satisfies for some The number is called the pursuit time.

Let us introduce some additional notation. Suppose that

(7)

(8)

where, are positive numbers,

System (2), where and satisfy condition (5) [respectively, (6)], will be referred to as the pursuit game (2), (5) [respectively, (2), (6)]. ). Suppose that

(9)

Then the solution of problem (2) is given by

(10)

where the functions satisfy the following infinite system of differential equations and initial conditions

(11)

The solution of the system (11) is given by

(12)

where,

3. Pursuit Differential Game

Theorem 2.1. If then in the pursuit game (2), (5) it is possible to complete the pursuit from any initial position

Proof: Let is an arbitrary control of the second player.

We use the control method proposed in ⁵, we shall seek the pursuers control as in the form

(13)

where, are constants and satisfies the following:

(14)

Define

(15)

If we set in (13), we get

(16)

Suppose that is an arbitrary point of the set . We now show that the control (16) ensure that for all .

By substituting the control (16) into the solution (12), we get

(17)

Hence, at

(18)

Therefore, in the pursuit game problem (2), (5), it is possible to complete the pursuit from the initial state for

Now, let us prove the admissibility the control (16). In the pursuit game (2), (5), the controls and satisfying inequalities (5). We have,

(19)

It follows that,

(20)

Let,

(21)

By setting on and taking into account (16), we see that

(22)

Applying the Minkowski inequality, we get

(23)

By using (20) and (21), we conclude that

(24)

Taking into account (7), we have

And finally, (21) implies that

From the above formula, we conclude that the control of the first player is admissible control. Thus, the pursuit in the pursuit game (2), (5), is completed from an arbitrary initial position . This complete the proof of theorem 1.

Theorem 2.2. If and , then in the pursuit game (2), (6), the pursuit is completed in the -neighborhood of zero (which means the solution ) of the pursuit game (2), (6), satisfies the inequality for some ) it is possible to complete the pursuit from any initial position

Proof: Consider the pursuit game (2), (6). Suppose that is an arbitrary point of the set is an arbitrary control of the second player.

Let be an arbitrary real number such that . We define the pursuer control by

(25)

where,

(26)

In accordance with (12), (25) and (26), we obtain

(27)

In view of (27), we have

(28)

But,

By using the cauchy-schwarz inequality, we conclude that

(29)

Since and it is easy to show that

(30)

From (29) and (30), it follows that

(31)

Since, this implies that

(32)

where,

(33)

Now, if , then by the definition the pursuit in the pursuit game (2), (6), it is possible to complete from the initial state for .

In the case , it follows from (32) that

(34)

We assume that and define the pursuers control by:

(35)

Moreover, according to (26) and (35), it is easy to see that

(36)

By a direct computation using (27), (28) and (36), we get

(37)

In the case . By the definition the pursuit in the pursuit game (2), (6) is completed from the initial state for . In the case

By setting and use the previous argument, we conclude that

(38)

where,

(39)

Now, we will prove that the pursuit in the pursuit game (2), (6) is complete from the initial state in finitely many steps.

First, let k be the smallest positive integer number such that

(40)

In the kth step, we have the following two cases:

1)

2)

In case 2), by setting where

Continuing and use in the previous arguments, we conclude that

(41)

where,

(42)

By using (34), (38) and (41), we obtain

It follows from (40) that

(43)

On the other hand by using (33),(39) and (42), we get

(44)

which contradicts the inequality (43), hence, the inequality does not hold. Therefore, the inequality in the case 1), holds.

Thus, by the definition it is possible to complete the pursuit (2), (6) from an initial position before the step at time at

Finally, we will estimate pursuit time. The above discussion shows that

Thus, we deduce from (34), (38) and (41), that

A simple calculation by using (40) yields

This ends the proof of the theorem.

4. Conclusion

By using a new control method, we have studied pursuit game problem with dynamics described by a partial differential equation of first order. We state and prove a theorem on pursuit with mixed constraints on control of players. Integral (geometric) constrain is imposed on the control of the pursuer whereas, that of the evader is subject to geometric (integral) constraint. In this theorem, we established the sufficient conditions for which pursuit is possible in the game considered.

References