OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov


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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS Supplement Volume 2005 pp OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics and Computer Sciences Texas Woman s University Denton, TX 76204, USA E.N. Khailov Department of Computer Mathematics and Cybernetics Moscow State Lomonosov University Moscow, , Russia Abstract. We consider a controlled system of differential equations modeling a firm that takes a loan in order to expand its production activities. The objective is to determine the optimal loan repayment schedule using the variables of the business current profitability, the bank s interest rate on the loan and the cost of reinvestment of capital. The portion of the annual profit which a firm returns to the bank and the value of the total loan taken by the firm are control parameters. We consider a linear production function and investigate the attainable sets for the system analytically and numerically. Optimal control problems are stated and their solutions are found using attainable sets. Attainable sets for different values of the parameters of the system are constructed with the use of a computer program written in MAPLE. Possible economic applications are discussed. 1. Introduction. Global competition and increasingly sophisticated business strategies are forcing businesses to make the most of every dollar. Profitable companies often take out commercial loans for expansion of production in order to remain at the forefront of their fields or in hopes of surpassing competitors. The commercial loans, unlike the individual loans, do not necessarily require monthly installments, giving to businesses an opportunity to decide the most advantageous schedule for repayment. From a mathematical point of view it is of interest to model the dynamics of such a situation and find the optimal loan repayment schedule. Let us consider a microeconomic model of a firm which takes a loan of value S at k% annual interest for T years. As a basis for our studies, we took the model created by Tokarev ([1],[2]) and investigated it respectively two control parameters: the portion of the profit which the firm decides to repay to the bank, u(t), and the total loan amount S, that we suggest the company borrows. Naturally, 0 u(t) 1 and 0 S S; here S is a maximum value of the credit. We assume that the entire loan S is taken at moment t = 0, and it is immediately transformed into production funds x 1. The firm repays the debt and the interest, which is added continuously using the compound interest formula under the fixed total interest rate k. The firm has the right to chose a loan repayment schedule; the only constraint is that the total debt must be completely paid off at the moment t = T. Then the current 2000 Mathematics Subject Classification. 49J15, 58E25, 90A16, 93B03. Key words and phrases. Controlled system, attainable set, microeconomic dynamical model. 345
2 346 E.V. GRIGORIEVA AND E.N. KHAILOV debt, x 2 (T ), becomes 0. The current net profit of the firm, P, can be determined by the production function F (x 1 ) with respect to the achieved level of production funds x 1 as P = rf (x 1 ), F (x 0 1) = x 0 1, F (x 1 ) > 0, F (x 1 ) 0. Here r (year 1 ) is the profitability of the firm with initial funds x 0 1. The firm is distributing the flow of its pure net profit P between the repayment to the bank ( up ) and refinancing its further development ( (1 u)p ). The proportion between these two expenses can be changed by the firm as a function of time u = u(t). Using the assumptions above, the dynamics of production funds x 1 (t) of the firm with the profit P and the debt x 2 (t) can be written as a nonlinear controlled system : ẋ 1 (t) = r(1 u(t))f (x 1 (t)), t [0, T ], ẋ 2 (t) = kx 2 (t) ru(t)f (x 1 (t)), x 1 (0) = x S, x 2 (0) = S, x 0 (1) 1 > 0, 0 u(t) 1, 0 S S. We assume that the firm s objective is maximizing its final production funds x 1 (T ), which maximizes the continuous profit, and that the debt of the firm must be completely paid off at time t = T, that is x 2 (T ) = 0. For a model with the linear production function, we investigate the attainable set and its properties for different parameters of the model. We prove that the attainable set is a compact convex set and that each point of the boundary of the attainable set can be reached by exactly one control taking values {0; 1} with at most one switching. Furthermore, we formulate the optimal control problems and look for their solutions using attainable sets and their properties. This approach of solving the optimal control problems differs our work from the works of Tokarev ([1],[2]). Moreover, the model is completely investigated analytically and numerically and can be used as a study tool at the business schools. 2. Model with Linear Production Function. The model (1) with the linear production function can be rewritten as ẋ 1 (t) = r(1 u(t))x 1 (t), t [0, T ], ẋ 2 (t) = kx 2 (t) ru(t)x 1 (t), (2) x 1 (0) = x S, x 2 (0) = S, x 0 1 > 0, S > 0, Here x 1, x 2 are variables, and r, k, S, x 0 1 and T are constant parameters; the function u(t) is the control function. Thus, the system (2) is a controlled bilinear system. We call D(T ) the control set, such that D(T ) is the set of all Lebesgue measurable functions u(t), satisfying the inequality 0 u(t) 1 for almost all t [0, T ]. The following lemma describes the property of the variable x 1 for system (2). Lemma 1. Let u( ) D(T ) be a control function. Then, the component x 1 (t) of the solution x(t) = (x 1 (t), x 2 (t)) of the system (2) corresponding to this control u(t), satisfies the inequality x 1 (t) > 0, t [0, T ]. (3) Here and further, the symbol means transpose. Proof is by direct integration of the first equation of the system (2).
3 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN An Attainable Set and its Properties. We denote X(T ) an attainable set for system (2) reachable from an initial condition x 0 = (x 0 1, x 0 2) at moment T ; that is X(T ) is the set of all ends x(t ) = (x 1 (T ), x 2 (T )) of trajectories x(t) = (x 1 (t), x 2 (t)) for system (2) under corresponding admissible controls u( ) D(T ). The set X(T ) is a compact set in R 2. ([3, pp ]). Then by Lemma 1 and work [4, Theorem 1], the following Theorem holds. Theorem 1. An attainable set X(T ) is a convex set in R 2. Now, we consider the case when r = k. By subtracting from the first equation of system (2) the second equation, we obtain d dt (x 1(t) x 2 (t)) = r(x 1 (t) x 2 (t)), t [0, T ]. Integrating this equality over the interval [0, T ], we have The following lemma holds. x 1 (T ) = x 2 (T ) + x 0 1e rt. (4) Lemma 2. If r = k, then the attainable set X(T ) is a segment in R 2. Now, we consider the general case r k. A point x on the boundary of the attainable set X(T ) corresponds to the control u ( ) D(T ) and to the trajectory x (t) = (x 1(t), x 2(t)), t [0, T ] of system (2), such that x = x (T ). Then it follows from [3, pp ] that there exists a nontrivial solution ψ(t) = (ψ 1 (t), ψ 2 (t)) of the adjoint system: ψ 1 (t) = r(1 u (t))ψ 1 (t) + ru (t)ψ 2 (t), ψ 2 (t) = kψ 2 (t), t [0, T ], ψ 1 (T ) = ψ 1, ψ 2 (T ) = ψ 2, for which 0, L(t) < 0, u (t) = [0, 1], L(t) = 0, (6) 1, L(t) > 0. Here L(t) = (ψ 1 (t) + ψ 2 (t)) is the switching function, which behavior determines the type of control u (t) leading to the boundary of the attainable set X(T ). The vector ψ = (ψ1, ψ2) 0 is a support vector to the set X(T ) at the point x. Using the adjoint system (5) and the definition of the switching function L(t), it is easy to see that the function L(t) satisfies the Cauchy problem : { L(t) = r(1 u (t))l(t) + (k r)ψ2e k(t t), t [0, T ], L(T ) = (ψ1 + ψ2). (7) The following lemma results from the analysis of the Cauchy problem (7). Lemma 3. The switching function L(t) is a nonzero solution of the Cauchy problem (7). Lemma 3 allows us to rewrite (6) in the following form : { 0, L(t) < 0, u (t) = (8) 1, L(t) > 0. At points of discontinuity we define the function u (t) by its left limit. Consequently, the control u (t), t [0, T ] leading to the point x on the boundary of the attainable set X(T ) is a piecewise constant function, taking values {0, 1}. (5)
4 348 E.V. GRIGORIEVA AND E.N. KHAILOV Next, we will determine the number of switchings of the control function u (t), t [0, T ]. It follows from (8) that it is sufficient to find the number of zeros of function L(t) on the interval (0, T ). Lemma 4. The switching function L(t) has at most one zero on the interval (0, T ). Proof. Let us assume that the function L(t) has at least two zeros τ 1, τ 2, such that τ 1 < τ 2 on the interval (0, T ). That is L(τ 1 ) = 0, L(τ 2 ) = 0. (9) Integrating the differential equation from (7) on the interval [τ 1, τ 2 ] and using the equalities (9), we obtain (k r)ψ 2 τ2 τ 1 e k(t s) e r s (1 u (ξ))dξ τ 1 ds = 0. The expressions under the integral are positive and k r. Therefore, ψ 2 = 0 necessarily holds to satisfy the equality. Then from the differential equation in (7) and relationships (9) it follows that L(t) = 0, on [0, T ], and the initial condition from (7) leads to the equality ψ 1 = 0. We obtain that ψ = (ψ 1, ψ 2) = 0. This fact contradicts the nontriviality of the vector ψ. Therefore, the function L(t) has at most one zero on the interval (0, T ). Based on the obtained results, using the inequality (3) from Lemma 1, Theorem 1, and the approach presented in [5, Theorem 6], we have the following statements. Theorem 2. Every point x on the boundary of the attainable set X(T ) associates to exactly one piecewise constant control u (t), t [0, T ], taking values {0; 1} and having at most one switching on (0, T ). Theorem 3. The attainable set X(T ) is strictly convex set in R The Problem of the Optimal Credit Repayment Plan. Now, for the system (2) and under the fixed value S > 0, we are to consider the first optimal control problem, J(u, S) = x 1 (T ) max, (10) u( ) D(T ) subject to x 2 (T ) = 0. We denote by J (S) the maximum value of the functional J(u, S). The optimal control problem (10) can be stated as : To find such optimal credit repayment schedule which results in the total debt paid off by the moment T and the total production maximized. It follows from (4) that for the situation when r = k the problem of optimization is not applicable. Therefore, further we will consider only the situation when r k. It follows from Section 3 that the boundary of the attainable set X(T ) for system (2) is the union of two oneparameter curves ξ(θ) = (ξ 1 (θ), ξ 2 (θ)), and χ(τ) = (χ 1 (τ), χ 2 (τ)), where θ and τ [0, T ]. The first curve, ξ(θ), consists of all ends x (T ) of trajectories x (t), t [0, T ] of the system (2), corresponding to controls { 1, 0 t θ, v (t) = (11) 0, θ < t T. The second curve, χ(τ), is formed by the values x (T ) of trajectories x (t), t [0, T ] of the system (2) corresponding to the controls { 0, 0 t (T τ), w (t) = (12) 1, (T τ) < t T.
5 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN 349 Here (T τ) and θ are the moments of switchings. Direct integration of the system (2) under the controls (11) and (12) gives us the following expressions for the components of the functions ξ(θ) and χ(τ) : ξ 1 (θ) = (x S)e r(t θ), θ [0, T ], ξ 2 (θ) = Se kt r k (x0 1 + S)e k(t θ) (e kθ 1), χ 1 (τ) = (x S)e r(t τ), τ [0, T ], χ 2 (τ) = Se kt r k (x0 1 + S)e r(t τ) (e kτ 1). (13) It is obvious that the functions ξ(θ) and χ(τ) satisfy ξ 1 (0) = χ 1 (0) = (x S)e rt, ξ 2 (0) = χ 2 (0) = Se kt, ξ 1 (T ) = χ 1 (T ) = x S, ξ 2 (T ) = χ 2 (T ) = Se kt r k (x0 1 + S)(e kt 1). (14) It follows from Section 3 that in order to solve the problem (10) it is sufficient to find the point with the largest value of the first component among the points of intersection of the curves ξ(θ) and χ(τ) with the x 1 axis. Therefore, the condition under which a solution of the optimal control problem (10) exists, is { } Q = min min ξ 2(θ), min χ 2(τ) < 0. (15) θ [0,T ] τ [0,T ] Now, we will analyze the left part of the inequality (15). We find the minimum of the function ξ 2 (θ) on the interval [0, T ]. Differentiating the second expression in (13) with respect to θ we obtain : ξ 2 (θ) = r(x S)e k(t θ) < 0. Hence the function ξ 2 (θ) monotonically decreases on the interval [0, T ] and min ξ 2(θ) = ξ 2 (T ). (16) θ [0,T ] Next, we investigate the minimum of the function χ 2 (τ) on the interval [0, T ]. Differentiating the fourth expression in (13) with respect to τ we obtain : χ 2 (τ) = r k (x0 1 + S)e r(t τ) (r + (k r)e kτ ). (17) To investigate the behavior of χ 2 (τ) we need to consider two cases. Case 1. If r < k, then χ 2 (τ) < 0 in (17). Hence the function χ 2 (τ) monotonically decreases on the interval [0, T ] and min χ 2(τ) = χ 2 (T ). (18) τ [0,T ] Case 2. If r > k, then ( it) follows from (17) that the function χ 2 (τ) reaches its minimum at τ 0 = 1 k ln r. r k r If r k > ekt, then τ 0 > T and the minimum of the function χ 2 (τ) on the interval [0, T ] is at τ = T. Therefore, the equality (18) is valid.
6 350 E.V. GRIGORIEVA AND E.N. KHAILOV r If r k ekt, then τ 0 (0, T ] and the function χ 2 (τ) reaches its minimum at an inner point τ 0 of the interval [0, T ] and then min χ 2(τ) = χ 2 (τ 0 ) χ 2 (T ). (19) τ [0,T ] Now, for analysis of the expression (15) we consider the following cases : (a) : let r < k; r (b) : let r > k and r k > ekt ; r (c) : let r > k and r k ekt. Next, we will study the cases (a)(c). In cases (a) and (b) from (14), (16), (18) we have the relationship : Q = min ξ 2(θ) = ξ 2 (T ) = χ 2 (T ) = min χ 2(τ). θ [0,T ] τ [0,T ] From (15) it follows that in cases (a) and (b) the solution of the optimal control problem (10) exists if the following inequality is valid : Se kt r k (x0 1 + S)(e kt 1) < 0. (20) In case (c) from (14), (16) and (19) we have the relationship : Q = min χ 2(τ) = χ 2 (τ 0 ) χ 2 (T ) = ξ 2 (T ) = min ξ 2(θ). τ [0,T ] θ [0,T ] It is easy to show that in this case the following inequalities hold : χ 2 (τ 0 ) < 0, χ 2 (T ) < 0. (21) Therefore, the inequality (15) is valid and the solution of the optimal control problem (10) exists. Further, we will find the solutions of the optimal control problem (10) in the cases (a)(c). We consider the cases (a) and (b). The relationships ξ 2 (0) = χ 2 (0) > 0 and ξ 2 (T ) = χ 2 (T ) < 0 follow from (14) and (20). Moreover, the derivatives ξ 2 (θ) < 0 and χ 2 (τ) < 0 for all θ, τ [0, T ]. This means that the curve ξ(θ) has precisely one point of intersection with x 1 axis as does the curve χ(τ). Thus, there exist the values θ, τ (0, T ), such that Moreover, the following equalities hold : ξ 1 (θ ) = (x S)e r(t θ ), ξ 2 (θ ) = 0, χ 2 (τ ) = 0. (22) ξ 2 (θ ) = Se kt r k (x0 1 + S)e k(t θ ) (e kθ 1) = 0, χ 1 (τ ) = (x S)e r(t τ ), χ 2 (τ ) = Se kt r k (x0 1 + S)e r(t τ ) (e kτ 1) = 0. Next, we compare the values ξ 1 (θ ) and χ 1 (τ ). We estimate the value From (23) we have the equality : χ 2 (θ ) = Se kt r k (x0 1 + S)e r(t θ ) (e kθ 1). χ 2 (θ ) ξ 2 (θ ) = r k (x0 1 + S)(e kθ 1)(e k(t θ ) e r(t θ ) ). (23)
7 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN 351 It follows from this relationship that in the case (a) the inequality χ 2 (θ ) > 0 is valid and in the case (b) the inequality χ 2 (θ ) < 0 holds. Hence in the case (a) we obtain χ 2 (θ ) > 0 = χ 2 (τ ). The function χ 2 (τ) decreases on the interval [0, T ] and therefore, we have the inequality θ < τ. In the case (b) we obtain χ 2 (θ ) < 0 = χ 2 (τ ). The function χ 2 (τ) decreases on the interval [0, T ] and thus, we have the inequality θ > τ. Then we consider the equality ξ 1 (θ ) χ 1 (τ ) = χ 1 (θ ) χ 1 (τ ). We find from this relationship and the previous arguments that in the case (a) the inequality ξ 1 (θ ) > χ 1 (τ ) is valid, and in the case (b) the inequality ξ 1 (θ ) < χ 1 (τ ) holds. Summarizing all previous arguments, we conclude that : in case (a) the maximum value of functional J(u, S) is J (S) = ξ 1 (θ ), where the expression ξ 1 (θ ) and the value θ are defined by the first and second expressions in (23). The corresponding optimal control u (t), t [0, T ] has the type (11) with the switching θ (0, T ); in case (b) the maximum value of functional J(u, S) is J (S) = χ 1 (τ ), where the expression χ 1 (τ ) and the value τ are defined by the third and fourth expressions in (23). The corresponding optimal control u (t), t [0, T ] has the type (12) with the switching (T τ ) (0, T ). In cases (a) and (b) the relationship (20) is always true. Next, we consider the case (c). In this case for the function ξ 2 (θ) it follows from (14) and (21) that ξ 2 (0) > 0 and ξ 2 (T ) < 0. Moreover, on the interval [0, T ] we have the inequality ξ 2 (θ) < 0. Hence the curve ξ(θ) has only one point of intersection with the x 1 axis. For the function χ 2 (τ) it follows from (14) and (21) that χ 2 (0) > 0 and χ 2 (τ 0 ) < 0. Moreover, we have the inequality χ 2 (τ) < 0 on the interval [0, τ 0 ). Hence the curve χ(τ) has only one point of the intersection with x 1 axis as well. Therefore, we define the values θ (0, T ) and τ (0, τ 0 ) such that the equalities (22) and (23) are valid. Then we compare the values ξ 1 (θ ) and χ 1 (τ ) by arguments similar to cases (a) and (b). We obtain the inequality ξ 1 (θ ) < χ 1 (τ ). Hence the conclusion in the case (c) is similar to the case (b). 5. The Problem of the Optimal Maximum Repayable Credit. In this section we want to find the value of the maximum repayable credit, which is equal to or less than the bank cutoff S (the value of the maximum credit offered by the bank) and the most advantageous for the firm. This can be written as the second optimal control problem : J(u, S) max u( ) D(T ),S [0, S] (24) subject to x 2 (T ) = 0. Using the results of Section 4 we can rewrite (24) as ( ) max J(u, S) = max u( ) D(T ),S [0, S] S [0, S] max J(u, S) u( ) D(T ) = max J (S). (25) S [0, S] It follows from (25) that we can use formulas for J (S) from Section 4. Also we denote by J the maximum value of the function J (S). We consider the cases (a)(c) as we did in Section 4. Now, we study case (a). It follows from (25) and results of Section 4 that it is sufficient to consider the problem : J (S) = (x S)e r(t θ (S)) max S [0, S] (26)
8 352 E.V. GRIGORIEVA AND E.N. KHAILOV subject to the condition (20), which transforms to the inequality S < rx0 1(e kt 1). (27) r + (k r)ekt Here the value θ (S) [0, T ) is the only root of the equation Se kt r k (x0 1 + S)e k(t θ (S)) (e kθ (S) 1) = 0 (28) for every value S [0, S]. We define the value Ŝ asŝ { rx = min S, 0 1(e kt } 1) r + (k r)e kt. (29) Then we express the value θ (S) from (28) and insert it into the expression (26) for the function J (S). Hence, we obtain instead of problem (26)(28) the following problem : J (S) = (x S)e rt (1 ) r k ks r(x S) max S [0,Ŝ]. (30) For the solution of the problem (30) we calculate the derivative of the function J (S) : J (S) = r k ( ) r k k ks ks k r(x S)erT r(x S). It is easy to see that J (S) < 0 for all S (0, Ŝ]. Then the function J (S) decreases on the interval (0, Ŝ]. Therefore, the maximum value of the function J (S) in (30) is J = J (0) and the corresponding value of S is S = 0. Hence it is not advantageous for the company to take any credit. Next, we study case (b). It follows from (25) and results of Section 4 that it is sufficient to consider the problem : J (S) = (x S)e r(t τ (S)) max S [0, S] subject to the condition (27), where τ (S) [0, T ) is the only root of the implicit equation (31) Se kt r k (x0 1 + S)e r(t τ (S)) (e kτ (S) 1) = 0 (32) for every value S [0, S]. Determining the value Ŝ by (29) we rewrite the problem (31), (32) as the following problem : J (S) = (x S)e r(t τ (S)) max, (33) S [0,Ŝ] where the value τ (S) is found from the equality (32) for every S [0, Ŝ]. For the solution of the problem (33) we calculate the derivative of the function J (S) : J (S) = e r(t τ (S)) (1 r(x S) τ (S)). (34) We find from (32) the derivative of the function τ (S) : τ (S) = kx 0 1e kt r(x S)2 e r(t τ (S)) ((k r)(e kτ (S) 1) + k).
9 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN 353 Then we insert it into the expression (34) and obtain : J (S) = k(er(t τ (S)) e k(t τ (S)) ) e kτ (S) ((k r)(e kτ (S) 1) + k). It is easy to see that J (S) > 0 for all S [0, Ŝ]. Thus, the function J (S) increases on the interval [0, Ŝ]. Therefore, the maximum value of the function J (S) in (33) is J = J (Ŝ) and the corresponding value of S is S = Ŝ. Hence the company must take the entire available credit Ŝ. Next, we study case (c). It follows from (25) and results of Section 4 that it is sufficient to consider the problem (31), (32). We use arguments similar to the arguments from case (b) to find the solution of the problem. The maximum value of the function J (S) is J = J ( S) and the corresponding value of S is S = S. Here again the company must take the entire available credit S Debt Production Figure 1. r = 0.1, k = 0.5, x 0 1 = 1, S = 1, T = Production Debt Figure 2. r = 0.5, k = 0.1, x 0 1 = 1, S = 50, T = 5.
10 354 E.V. GRIGORIEVA AND E.N. KHAILOV 6. Computer Modeling. Knowing how to get to the boundary of an attainable set, we wrote a computer program in MAPLE that builds the attainable set X(T ) as the region in R 2, bounded by the union of two oneparameter curves, ξ(θ) and χ(τ). The program solves the Cauchy problems (2) for bangbang controls v(t) defined by (11) and w(t), defined by (12). The moments of switching θ and (T τ) change within the interval [0, T ]. Both curves are displayed on the same graph forming the boundary of the attainable set X(T ). The curve ξ(θ) is shown as a thin dashed curve and χ(τ) as a thick solid curve. Attainable sets X(T ) for the cases (a), (c) are shown on Figures 1 and 2, respectively. As we proved in Section 3, all attainable sets X(T ) are strictly convex. For the first optimal control problem (10) the maximum value of the functional and the switching of the corresponding optimal control are obtained analytically. Figure 1 represents the situation when r = 0.1, k = 0.5, x 0 1 = 1, S = 1, and T = 5. The maximum value of the functional J(u, S) is J (S) = 3.06 and comes from control of type v(t) with the moment of switching θ = It looks like the firm must pay the debt during the first 3.6 years and then spend the remaining time for development. However, it follows from Section 5 that, if r < k, then it is not beneficial for a firm to take any credit from the bank. This fact completely agrees with any book on financial management ([6]). The attainable set in Figure 2 is constructed for the following parameters : r = 0.5, k = 0.1, x 0 1 = 1, S = 50, and T = 5. Since r > k, then the maximum value of the functional J(u, S) is J (S) = and comes from control of type w(t) with the switching at (T τ) = 4.7. Therefore, in order to maximize production and eliminate the debt the company must take the maximum possible credit from the bank and first develop during 4.7 years and then spend the remaining time to pay the debt. This work is supported TWU REP Grant, RFFI Grant and RFFI Support Grant of the Leading Scientific Schools SS REFERENCES [1] V.V. Tokarev, Unimproving extension and the structure of extremals in control of the credit, Automation and Remote Control, 62 (2001) no. 9, [2] V.V. Tokarev, Optimal and admissible programs of credit control, Automation and Remote Control, 63(2002), no. 1, [3] E.B. Lee, L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, [4] R.W. Brockett, On the reachable set for bilinear systems, Lecture Notes in Economics and Mathematical Systems, 111 (1975), [5] O. Hajek, Bilinear control : rankone inputs, Funkcialaj Ekvacioj, 34 (1991), [6] A. Allen, Financial Risk Management, John Wiley & Sons, New York, Received September, 2004; revised March, address: address:
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