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In this article, affine-quadratic control problems are studied. Error bounds are derived for the difference between the performance indices corresponding to the optimal and a class of suboptimal controls. In particular, it is shown that the performance of these suboptimal controls is close to that of the optimal control whenever the error in estimating the costate initial condition is small.

One of the most active areas in control theory is optimal control and methods to find them [

In this article, we study the affine-quadratic control problem given by ((1), (2)). We note that a method for finding the initial condition for the costate is recently proposed [

The article is organized as follows. In Section 2, the affine-quadratic control problem is described. We also explain how to obtain the optimal control in terms of costate. The main (Theorem 2) is proved in Section 3. This theorem provides a method to obtain the costate (without the knowledge of its terminal value) which results in an explicit formula and performance bounds for a class of suboptimal controls.

Notation: For

We consider the affine control system

with the quadratic cost functional

Here

Throughout this paper, it is assumed that

from the admissible control space

Under these assumptions, for each admissible control

the control system (1) denoted by

The value function of the control problem given by (1), (2), is defined as

A control input

Similarly a control input

Given

where

To derive an expression for the optimal control

system:

Here

control system (1), (2), which provides a set of necessary conditions for

Theorem 1 [PMP] Let

corresponding to

for

attains minimum at

Corollary 1 Let

corresponding to

Proof. The proof follows immediately from the above theorem. □

Now to obtain

together with the initial conditions

In general, solving this coupled system and finding a closed form solution

it may be easier to find

difference between the performance indices corresponding to

In this section, we prove the main result.

Theorem 2 Consider the affine-quadratic control problem (1), (2). Let

control as given in (5),

Also let

initial condition

where

The constant

Proof. Note that

(6)

From R.H.S. of (6), we first consider the term

By adding and subtracting

Therefore

From R.H.S. of (6), we next consider the term

In a similar manner (as for (7)), we have

From R.H.S. of (6), we next consider the term

Let us have

In the above term, put the

Now using assumption on the matrix function

Using this and following the procedure as for the inequality (7), we get

Therefore

Hence the result follows by the inequalities (7), (8), and (9). □

Remark 3 It follows from the previous theorem that

This implies that