One of the most active areas in control theory is optimal control and methods to find them [1] - [3] . It has a wide range of practical applications in engineering (Aerospace, Chemical, Mechanical, Electrical), science (Physics, Biology), and economics (see e.g. [4] - [7] ). Optimal control theory has been developed for linear systems ( [1] [2] [8] ) and explicit formulae for computing optimal control inputs are available. However, control of nonlinear systems is much more challenging and obtaining formulae for optimal controls seems in general not possible. This motivated researchers to study various classes of nonlinear control problems separately, and affine-qudratic problems is one such class. In a recent paper [9] , the optimal control for affine-quadratic problems is obtained in terms of the associated costate. But, in practice, it is difficult to compute the costate (at each time) as the knowledge of its terminal condition is required.

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 [10] . This allows one to compute the initial costate (at) exactly or approximately. This approximation of the initial costate and the explicit formula for optimal control (as in [11] ) are shown, in this article, which give rise to suboptimal controls of practical importance. More precisely, our main theorem (Theorem 2) provides an upper bound for the difference in performance between these suboptimal and optimal control.

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, , and, we use the notation

, ,.

We consider the affine control system

with the quadratic cost functional

Here is the state vector, is the control vector, ,

, , , , and ' denotes transposition.

Throughout this paper, it is assumed that are positive semidefinite, is positive definite, the functions are continuously differentiable with bounded derivatives, the control input is chosen

from the admissible control space.

Under these assumptions, for each admissible control there exist a unique solution (trajectory) of

the control system (1) denoted by.

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

.

A control input is optimal (for) if

.

Similarly a control input is -optimal (for) if

.

Given, the optimal control problem is to find a control which minimizes the cost functional

. The Hamiltonian associated with the optimal control problem (1), (2), is given as

where is the adjoint vector.

To derive an expression for the optimal control (for), it is convenient to introduce the adjoint

system:

Here. We now state the Pontryagin’s Minimum Principle (PMP) for the affine-quadratic

control system (1), (2), which provides a set of necessary conditions for to be optimal [12] .

Theorem 1 [PMP] Let, , and. Also let be the adjoint vector

corresponding to and, as given by the equation (4). Then for a control input to be optimal

for, it is necessary that the map

,

attains minimum at, for a.e..

Corollary 1 Let, , and. Also let be the adjoint vector

corresponding to and, as given by the equation (4). Then the optimal control (for) is

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

Now to obtain (in (5)) in terms of, we solve the coupled systems given in (1) and (4)

together with the initial conditions and respectively.

In general, solving this coupled system and finding a closed form solution is very difficult. However

it may be easier to find approximately. Such an approximation will lead to the associated adjoint state

and admissible control. In the next section, we provide bounds for the

difference between the performance indices corresponding to and.

In this section, we prove the main result.

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

control as given in (5), , and be the adjoint vector corresponding to and.

Also let be a suboptimal control and be the solution of the coupled system ((1), (4)) with

initial condition. Then

,

where

The constant depends only on the matrix function and the constant depends only on its gra- dient.

Proof. Note that

(6)

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

.

By adding and subtracting inside the integral, we get

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 matrix as and the matrix

as for each. Then we have,

Now using assumption on the matrix function, we have that the matrix function is continuously differentiable and has bounded derivatives. Therefore

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, when.

This implies that is a good suboptimal control when is a good approximation of

. We emphasize the fact that (and hence) can be computed at each time as is known.