OPTIMAL CONTROL OF LOGISTIC BIOECONOMIC MODEL WITH SINGULARITY-INDUCED BIFURCATION

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 5, Number 3-4, Pages c 9 Institute for Scientific Computing and Information OPTIMAL CONTROL OF LOGISTIC BIOECONOMIC MODEL WITH SINGULARITY-INDUCED BIFURCATION YUE ZHANG, QINGLING ZHANG Abstract. In this paper, the problems of singularity-induced bifurcation and impulsive behavior for a class of bioeconomic model are investigated by the theory of differential-algebraic equation. The singularity-induced bifurcation and impulsive behavior are found when the revenue gross is equal to the cost. An optimal controller is designed to eliminate the singularity-induced bifurcation and impulsive behavior by controlling the fishing effort. It causes the system to stabilize and come into an ideal state, thus not only protecting the ecologic source but also retrieving the bioeconomic system. Finally, numerical simulations are given to illustrate that the optimal controller is effective. Key Words. Bioeconomic model, Singularity-induced bifurcation, Impulsive behavior, Optimal control. 1. Introduction Recently, there has been a great interest shown in applications of differentialalgebraic equations DAEs to the analysis of some practical systems. Especially, various bifurcations can be found in DAE models [1, ]. The bifurcations in DAEs include, the typical saddle-node bifurcations, Hopf bifurcations, focus-node bifurcations, and the singularity-induced bifurcations SIBs. The SIBs occur when an equilibrium point is placed at a singularity manifold of the algebraic subsystem. The presence of the singularity manifold in DAEs is a challenging phenomenon and results in certain types of behavior that are not present in systems described by ordinary differential equations ODEs. Papers [3 5] present interesting results regarding the flow of DAEs near a singular equilibrium. At the same time, there have been also many results about impulsive behavior of DAEs [6, 7] and a great interest shown in applications of DAEs to the analysis of chemical processes [7] and electric power systems [8]. However, there has been few research papers describing DAEs models for bioeconomy, which may reflect many particular bioeconomic phenomena. Especially, the research on the singularity-induced bifurcations and impulsive behavior in bioeconomic system is seldom done. The SIB and impulsive behavior may change the stability properties of the associated operating points. Broadly speaking, a SIB describes a phenomenon of spectral divergence through infinity when an equilibrium locus of a DAE undergoes a singularity. In engineering systems, impulses may cause degradation in performance, damage components, or even destroy the system. Therefore, they are undesirable and one should eliminate the SIB and impulsive behavior through appropriate control methods. Received by the editors July 11, 8 and, in revised form, October 1, 8. Mathematics Subject Classification. 35R35, 49J4, 6G4. This research was supported by National Natural Science Foundation of China

2 37 Y. ZHANG, Q. L. ZHANG In recent years, much research has been devoted to the problem of harvesting a renewable resource in the most profitable manner. For example, the problems of maximizing yield, of preventing extinction of the population or both or bifurcation are investigated in papers [9 11]; the effect of impulsive perturbation on the growth of population and the optimal harvesting policies for periodic logistic equation with impulsive harvest are discussed in paper [1]. In these papers, the authors mainly concentrate on the case that the profit is not zero and don t discuss the case that the profit is zero. In fact, the zero profit case is inevitable in the bioeconomic course and its research is often ignored. However, there exist some important phenomena on this case, for example, bifurcation and impulsive behavior, which may affect the stability of the system, or even destroy the system. In this status, the population may grow explosively or become extinct instantly. So, it is significant to investigate the dynamic behavior of the system on the zero profit case and control it for avoiding the biology tragedy, protecting the biology system, retrieve the bioeconomic system and so on. In this paper, we will presented some results for a class of bioeconomic model on the zero profit case. The results on the zero profit case complement the significant results of papers [9 1]on the case that the profit is not zero. The following system is considered 1 ẋt = rxt1 xt k Ext xt is the density of population at time t, r is the intrinsic growth of population, k is the carrying capacity of environment, E is the fishery effort. In 1954, Gordon found the theory of open-access or public fishery and presented that the revenue gross subtracts the cost is the profit or the surplus production, i.e., T R T C = m [9], T R is the sustainable revenue gross, T R = py E, p is the price of a unit of harvested biomass, Y E is the sustainable yield; T C is the overall costs, T C = ce, c is the cost of a unit of effort, m is the profit or the surplus production. Based on system 1 and the above idea, when the effort Et changes by the time t, we have the following model { ẋt = rxt1 xt K Etxt = Etpxt c m the meanings of xt, r, k are same as 1, Et is the fishery effort at time t. In the following sections, we will investigate the dynamic behavior of the system on the zero profit case. Here, we choose to make the profiti.e. m a constant. Moreover, m is a bifurcation parameter in the following discussion. petxt is the sustainable revenue gross, cet is the overall costs. The model can be explained the exploitation for open access resources, for example, open access fishery and other wilding natural resources. The algebraic equation in system represents the relations of the revenue gross, the cost and the profit. A smooth solution of system may exist when the conditions that satisfy the constraint Et pxt c m =, pxt c. In this case, the DAE can be transformed an ODE with the same solution, to which we may apply an ODE theory. The main purpose of the paper is to show that there exists the SIB and the impulsive behavior in system when the profit is zero, which may change the stability, or even destroy the whole bioeconomic system. In this status, the magnitude of the population changes unexpectedly, for example, the fish population may suddenly grows explosively because one doesn t harvest the population when the profit is zero. It is significant to concentrate on the zero profit case in order to

3 OPTIMAL CONTROL OF LOGISTIC BIOECONOMIC MODEL WITH SIB 371 protect the biology source and retrieve the bioeconomic system. Furthermore, the impulsive behavior is eliminated by the controller which stabilizes the bioeconomic system and minimizes the cost energy, besides one may obtain some profit due to the costs decrease by the instrument of the government. The paper is organized as follows. In Section, a few basic properties of matrix pencils and the basic concepts of SIB in DAEs are introduced. In Section 3, the stability of equilibria in system is analyzed when m =. In Section 4, the analysis of SIB is given and the impulsive behavior comes into being in system when m =. The optimal controller which makes the system impulse free is designed when m = in Section 5. In Section 6, numerical simulations show that the controller is effective. Finally, concluding remarks are given in Section 7.. Matrix pencil and SIB in DAEs.1. Matrix pencil and Index. Given two n n matrices  and ˆB, the matrix pencil Â, ˆB is defined as the family {s C : sâ + ˆB}. The spectrum σâ, ˆB of the pencil is σâ, ˆB = {s C : detsâ ˆB = }. The matrix pencil is said to be regular if there is some s C such that detsâ ˆB. In this status, it may be reduced to the so-called Weierstrass normal form. Namely, it may be summarized as the following result. Lemma 1. [13] Suppose that Â, ˆB is a regular matrix pencil on R n. One can write R n = U V and find nonsingular maps P LR n and Q LU V, R n such that P ÂQ = In q and P N ˆBQ Ĉ =, there is an integer I q q 1, called the index of Â, ˆB and denoted by indâ, ˆB, satisfying N q = and N q 1. Now Ĉ, I n q LU, N, I q LV and I n q and I q are identities on U and V, respectively. Moreover, σâ, ˆB = σĉ and #σâ, ˆB = dimu, the symbol # denotes cardinality... SIB in DAEs. For differential-algebraic equations DAEs { ẋ = fx, y, µ 3 = gx, y, µ f : R n1+n+1 R n1, g : R n1+n+1 R n, x X R n1, y Y R n, µ Λ R, define the set of all equilibria to be EQ and let OP denote the set of all stable equilibria defined as EQ = {x, y, µ X Y Λ : fx, y, µ =, gx, y, µ = }, OP = {x, y, µ EQ : detd y g, ReλJ <, J = D x f D y fd y g 1 D x g}. For a DAE operating away from the singularity manifold S {x, y, µ X Y Λ : detd y g = }, one may find saddle-node bifurcations, focus-node bifurcations, Hopf bifurcations and so on. SIB occurs if an equilibrium crosses the singular manifold S at the bifurcation point. Trajectories cross the singularity in a finite time with an infinite speed and the system changes stability due to an eigenvalue diverging to infinity. This type of bifurcation can be analyzed with the help of the following definition and theorem.

4 37 Y. ZHANG, Q. L. ZHANG Definition 1. [5]Singular Point Suppose that x, y, µ S, then x, y, µ is called a singular point of system 3. DAEs characterized by a local index change, have been the focus of considerable recent research. Singular equilibria in DAEs pose interesting bifurcation problems. in particular, the phenomenon known as the SIB not displayed in ODEs, has been originally characterized for DAEs by V.Venkatasubramanian et al. [3] Lemma. [3]Singularity Induced Bifurcation Theorem Consider system 3, suppose the following conditions are satisfied at x, y, µ : I fx, y, µ =, gx, y, µ =, D y g has a simple zero eigenvalue and trace[d y fadjd y gd x g] is nonzero. Dx f D II y f is nonsingular. D x g D y g D x f D y f D µ f III D x g D y g D µ g is nonsingular, D x D y D µ x, y, µ = detd y gx, y, µ. Then there exists a smooth curve of equilibria in R n1+n+1 which passes through x, y, µ and is transversal to the singular surface at x, y, µ. When µ increases through µ, one eigenvalue of the system i.e., an eigenvalue of J = D x f D y fd y g 1 D x g evaluated along the equilibrium locus moves from C to C + if B C > respectively, from C+ to C if B C < along the real axis by diverging through infinity. The other n 1 1 eigenvalues remain bounded and stay away from the origin. The constant B and C can be computed by evaluating at x, y, µ. B = trace[d y fadjd y gd x g], C = D µ D x D y D x f D y f D x g D y g 3. Stability of Equilibria 1 Dµ f D µ g In this section, we analyze the stability of equilibria of system when m =. Theorem 1. If m = and c >, then there exists three equilibria at P,, P1 k, and P c p, r1 c in system. Moreover, P, is locally unstable; P1 k, is locally asymptotically stable. Proof. Here, system has three equilibria at P,, P1 k, and P c p, r1 c provided that c >. The stability of equilibria depends on the eigenvalues of Jacobian matrix of system. It is given by r r J P = k x E x r pe px c P = c at P,. Let The eigenvalue of J P Ξ = 1. is given by the root of the polynomial λ r det =. c,

5 OPTIMAL CONTROL OF LOGISTIC BIOECONOMIC MODEL WITH SIB 373 So, the equilibrium P, is locally unstable. Similar to the discussion of the case of P,, the equilibrium P1 k, is locally asymptotically stable. This completes the proof. Nevertheless, the locally dynamic characteristic of the equilibrium P c p, r1 c is complicated that its characteristic equation becomes violated. This case will be analyzed in the next section. 4. SIB in bioeconomic system Since m is a bifurcation parameter, then the equilibrium P c p, r1 c coincides with the singular point, resulting in SIB when m =. In the following, we give the result. Theorem. If m = and c >, then system undergoes a SIB at equilibrium P c p, r1 c. Moreover, when m increases through m =, there is indeed a loss of stability for system near equilibrium P c p, r1 c. Proof. It is easily seen that equilibrium P c p, r1 c coincides with the singular point. Moreover, the following conditions are satisfied. I From detd E g P =, we can know that dim kerd E g P = 1. II traced x fadjd E gd x g P = trace x pe P = cr1 c. rc III f,g x,y P = c p r c k is nonsingular. IV detd E gxm, Em, m when m, xm, Em with P x, E denotes the smooth equilibrium path of system. By computing, we have B = traced x fadjd E gd x g P C = D m P D x D E D x f D E f D x g D E g = kp r c, = cr1 c, 1 Dm f D m g P = detd E g, and B C = cr c p k >. Based on Lemma and Theorem 1.1 in [4], the eigenvalue of system moves from C to C + along the real axis by diverging through infinity. Fig.1 shows that the eigenvalue λm diverges through infinity as m diverges zero, and λm has negative real part when m < ; λm has positive real part when m >. Moreover, when m increases through m =, there is indeed a loss of stability for system near the equilibrium P c p, r1 c. This completes the proof. Remark 1. Based on Lemma 1, the linearization of system along positive equilibrium locus yields index 1 matrix pencil Ξ, J for m and index matrix pencil Ξ, J for m =. Therefore, the index jumps by one at bifurcation point m =. Bioeconomic system exhibits an impulsive behavior when m =. Where 1 f, g r r Ξ =, J = x, E = k x E x. pe px c

6 374 Y. ZHANG, Q. L. ZHANG Fig.1 Qualitative representation of the movement of the eigenvalue near the singularity induced bifurcation of system Remark. If people harvest the population, the eigenvalue of bioeconomic system will diverge through infinity when the profit or the surplus production is zero, i.e., the revenue is equal to the cost. The index of the system is. Therefore, the system exhibits an impulsive behavior [7]. In this status, the magnitude of the population changes unexpectedly. This is one of characteristic of the paper. As a result, the bioeconomic system has a loss of balance. 5. Optimal Control for SIB in Bioeconomic system From the above result, we know that the index of matrix pencil Ξ, J is and jumps by one at the bifurcation point m = i.e. the profit is zero. So, there exist the SIB and impulsive behavior which is undesirable. One should make the index come to 1 in order to eliminate the SIB and impulsive behavior by designing a controller on the zero profit case. In the following, an optimal controller is designed. Here, we only consider the problem of cost energy minimum because we design the controller on the case of m =, which means that the profit is zero. Simply speaking, the function of the controller is to make the system stabilize and minimize the cost energy on the zero profit case. Consider the optimal control problem of the system 4 ΞẊt = fxt + gxtut, Xt = xt, Et T, rxt1 xt fxt = k Etxt Etpxt c 1 gxt =, 1 ut R is control input. The optimal control problem can be analyzed with the help of the following definition.,

7 OPTIMAL CONTROL OF LOGISTIC BIOECONOMIC MODEL WITH SIB 375 Definition. The optimal control problem of system 4, which is regular and rankξ < n, is to determine control u t such that the cost functional 5 Ω = u tdt is minimized, while eliminating the impulsive behavior and making the bioeconomic system stabilize. For the sake of convenience, let xt = xt x, Ẽt = Et E, x = c p, E = r1 c. We can transform system into the following system { xt = rc 6 = rp1 c xt pẽt c r k x t xtẽt xt + p xtẽt Then, the optimal control problem for system 4 is changed into the optimal control problem of the system 7 Ξ Xt = f Xt + g Xtũt, Xt = xt, ẼtT, f Xt rc = xt pẽt c r k x t xtẽt rp1 c, xt + p xtẽt g Xt 1 =, 1 ũt R is control input, which minimizes the cost functional 8 Ω = ũ tdt. We are looking for a state feedback ũt = H Xt + ṽt, H = h that locally minimizes cost function 8, h >. Consider the linearization of system 7 at P, 9 Ξ Xt = A Xt + B ũt, A = rc c p r c k, B =, ṽt = ṽ 1 t, T, 1 1 It is obvious that system 9 is impulsive controllable [14]. Putting ũt = H Xt + ṽt into system 9, we obtain that 1 Ξ Xt = à Xt + B ṽ 1 t, rc c p à =, h B = r c k 1..

8 376 Y. ZHANG, Q. L. ZHANG The index is 1. Thus, the optimal control problem of system 9 is changed into the problem of finding the controller ṽ 1 t of system 1 such that the cost function 11 Ω = ũ tdt = [ṽ1t + h Ẽ t]dt is minimal. By applying the quadratic regulation problem for singular systems, the optimal control of system 1 minimizing the cost function 11 is given by ṽ 1t = B T M Xt, M is the admittable solution to the following singular Riccati equations [15] Ã T M + M T Ã M T B BT M + h =, h Ξ T M = M T Ξ. By computing, we obtain that M M = c M + rp h 1 c rp h 1 c h, M = rc h 1 c rc + [ rc h 1 c rc ] + [rp1 c ], then 1 ṽ 1t = M xt. Putting 1 into ũt = H Xt + ṽt, we obtain the optimal control of system 9 13 ũ M xt t = h Ẽt and the optimal trajectory X opt { x 14 optt = rc + M x optt c pẽ optt = rp1 c x optt + hẽ optt Moreover, direct computation shows that system 14 is stable and no impulse terms exist in optimal trajectory X. To sum up, we obtain the locally optimal control u t of system 4 15 u M xt x t = h Et E and the optimal trajectory Xopt which satisfies { ẋ 16 optt = rc + Mx optt x c p E optt E = rp1 c x optt x + heoptt E The optimal cost function Ω is 17 Ω = M[x opt x ]. Based on the above discussion, the following result is obtained. Proposition 1. For system 4, the locally optimal control u t at the equilibrium P c p, r1 c is u t, which minimizes cost function 5. Moreover, optimal trajectory Xopt satisfies 16, h >.

9 OPTIMAL CONTROL OF LOGISTIC BIOECONOMIC MODEL WITH SIB 377 Remark 3. From 17, we know that the minimized cost function is decreasing as h increases, nevertheless, the time to eliminate the impulsive behavior and make the system stabilize is increasing see Fig. and Fig.3. Remark 4. Using the controller u t, we can eliminate the SIB and impulsive behavior in the bioeconomic system. Moreover, it can make the system stabilize. This may be completed by controlling the fishery effort, sequentially controlling the harvest for the population. The optimal control is mainly implemented through the instrument of the government. For the case of fishing, the enthusiasm of fishermen may be adjusted by all kinds of effort, for example, the tax may be increased or decreased, accordingly the costs may be changed. The aim is to avoid the biology tragedy, protect the fishery source, retrieve the bioeconomic system and so on. 6. Numerical simulations In this section, we examine the results of the numerical solution of system 4 with optimal controller 15 for certain values of the parameters. Our parameter values are based on Pacific halibut data over years and come from a variety of sources. We use estimates of r =.71/year and k = Kg see [9]. The price and cost data are based on the H. S. Mohring s reports 1973 in paper [9] that the ratio of the cost and the price c : p = Let p =.5/Kg, c = /year. In this status, system has an unique positive equilibrium P , and the optimal controller u t 18, when m =. u t = 18 [ 486 h h ] h xt Et By the optimal controller u t, system 4 is stable and come into the unique positive equilibrium P. It is shown that system 4 is stable and impulsive free in neighborhood of positive equilibrium P , in Fig. and Fig.3. Moreover, it can be seen that the waiting time to achieve control is increasing as h increases. In Fig., the controlled system can be quickly stabilized to the ideal state, i.e., the positive equilibrium P when h = 3. The time that the system is in shaky state is very short before the system is stabilized to the positive equilibrium P. The system approximately becomes a stable system nearby the positive equilibrium P, in other words, the population approximately remains at a constant level. However, in Fig.3, the time to make the system stabilize to the ideal state when h = is more than that when h = Conclusions In this paper, we study the problem of singularity-induced bifurcation and impulsive behavior for a class of bioeconomic model by the theory of differential-algebraic equation. Singularity-induced bifurcation is found when the revenue gross is equal to the costs. In this status, the impulsive behavior emerges. There exists the phenomenon that the magnitude of the population increases unexpectedly, which causes the ecosystem to become unbalanced. This unbalanced phenomenon can be eliminated by an optimal control policy. It confines the population change and make the population stabilize and come into an ideal state by controlling the fishery

10 378 Y. ZHANG, Q. L. ZHANG 1.76 x xt t a 1.5 Et t b Fig. The dynamic response of the controlled system with r =.71, k = , p =.5, c = , h = 3. a for xt population; b for effort Et x xt t a x Et t b x 1 6 Fig.3 The dynamic response of the controlled system with r =.71, k = , p =.5, c = , h =. a for xt population; b for effort Et.

11 OPTIMAL CONTROL OF LOGISTIC BIOECONOMIC MODEL WITH SIB 379 effort, thus not only protecting the ecologic source but also retrieving the bioeconomic system. For the case of fishing, the enthusiasm of fishermen may be adjusted by all kinds of effort for the sake of farsighted benefits, for example, the tax may be increased or decreased. From numerical simulations, it s shown that the bioeconomic system with SIB and impulsive behavior can be stable in neighborhood of an ideal state by the optimal controller. Acknowledgments The author thanks the anonymous authors whose work largely constitutes this sample file. This research was supported by National Natural Science Foundation of China References [1] S. Ayasun, C.O. Nwankpa and H.G. Kwatny, Computation of Singular and Singularity Induced Bifurcation Points of Differential-Algebraic Power System Model, IEEE Transactions on Circuits and Systems-I: Regular Papers, [] W. Marszalek, Z.W. Trzaska, Singularity-Induced Bifurcations in Electrical Power Systems, IEEE Transactions on Power Systems, [3] V. Venkatasubramanian, H. Schättler and J. Zaborszky, Local Bifurcations and Feasibility Regions in Differential-Algebraic Systems, IEEE Transaction on Automatic Control, [4] L.J. Yang, Y. Tang, An Improved Version of the Singularity-Induced Bifurcation Theorem, IEEE Transaction on Automatic Control, [5] R.E. Beardmore, The singularity-induced bifurcation and its Kronecker normal form, SIAM Journal of Matrix Analysis, [6] Q.L. Zhang, J. Lam, Robust impulse eleminating control for descriptor systems,dynamics of Continuous,Discrete and Impulsive Systems, Series B, [7] A. Kumar A., Prodromos Daoutidis. Control of Nonlinear Differential Algebraic Equation Systems-with Application to Chemical Processes, London. CRC Press LLC. 1999: [8] M. Hou, Technical notes and correspondence-controllability and elimination of impulsive modes in descriptor systems, IEEE Transactions on Automatic Control, [9] C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 199: [1] I. Foroni, L. Gardini, Rosser J.B. and Jr., Adaptive and Statistical Expectations in Renewable Resource Market, Mathematics and Computers in Simulation, [11] K.R. Fister, S. Lenhart, Optimal Control of a Competitive System with Age-structure, Journal of Mathematical Analysis and Applications, [1] Y.N. Xiao, D.Z. Cheng and H.S. Qin,Optimal Impulsive Control in Periodic Ecosystem, Systems and Control Letters, [13] F.R. Gantmacher, The Theory of Matrices, New York, Chelsea, [14] L.Y.Dai, Singular Control Systems, Heidelberg. Springer-Verlag Berlin. 1989: [15] D.M. Yang, Q.L. Zhang, B. Yao, Singular Systems, Beijing, Science Press, 4: the Institute of Systems Science, Northeastern University, 114, Shenyang, Liaoning, China zhangyue neu@sohu.com and qlzhang@mail.neu.edu.cn