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Research article

Optimal control problems with time inconsistency

  • Received: 15 September 2022 Revised: 04 November 2022 Accepted: 10 November 2022 Published: 11 November 2022
  • In the present study, the necessary and sufficient conditions of equilibrium control for general optimal control problems with time inconsistency are established in sense of open-loop. As an application, the linear quadratic optimal control problems with time inconsistency were also explored and an explicit equilibrium control is constructed.

    Citation: Wei Ji. Optimal control problems with time inconsistency[J]. Electronic Research Archive, 2023, 31(1): 492-508. doi: 10.3934/era.2023024

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  • In the present study, the necessary and sufficient conditions of equilibrium control for general optimal control problems with time inconsistency are established in sense of open-loop. As an application, the linear quadratic optimal control problems with time inconsistency were also explored and an explicit equilibrium control is constructed.



    Bellman dynamic programming principle [1] and Pontryagin maximum principle [2] serve as two of the most important tools in solving optimal control problems with time consistency (i.e., classical optimal control problems). However, as times goes by, the cost functional and control systems of optimal control problems with time inconsistency is changing, which makes these two methods are not suitable for such problems. Thus, to explore optimal control problems with time inconsistency, we adopt a game theoretic approach. The notion of "equilibrium" are therefore considered instead of "optimization".

    Actually, the research on time-inconsistent problems has a long history. The study of qualitative analysis on time-inconsistent behavior was carried out by Hume [3] in 1739 and by Smith [4] in 1759. These were then alluded to by Malthus in 1828, Pareto [5] in 1909, and Samuelson [6] in 1937. But until 1955, Strotz [7] in his milestone paper presented the mathematical formulation of the time-inconsistent problems. After that, the research on time-inconsistent problems is mainly divided into empirical research and theoretical research. The classical literature of empirical research includes the endowment theory of Thaler [8], the dynamic inconsistent theory of Kydland and Prescott [9] and the prospect theory of Kahneman and Tversky [10], who were Nobel Prize winners of economics in 2017, 2004 and 2002, respectively.

    In the literature on optimal control problems with time inconsistency, a large number of mathematicians and behavioral finance scientists have carried out research on theoretical research and obtained rich research results. The authors first introduced the definition of feedback equilibrium control in [11] and characterized differentiable sub-game perfect equilibria in a continuous time inter-temporal decision optimization problem with non-constant discounting function. In Bj¨ork and Murgoci [12], the authors generalized the extended HJB equation for a class of general controlled Markov process and a fairly general cost functional. They also proved that for every time-inconsistent optimal control problem, there is an associated time-consistent optimal control problem such that the optimal control for the time-consistent problem coincides with the equilibrium control for the time-inconsistent optimal control problem. In particular, we refer the reader to [13,14] and the references therein. In the series of works carried out by Yong and cooperative authors [16,17,18,19], the authors performed an alternative method by partition of time interval and then regarded the time-inconsistent problem as the limit of the time-consistent problem, and achieved a number of research results. Motivated from optimal control problems with time inconsistency, Hamaguchi [20] researched the local solvability of a flow of forward-backward stochastic differential equations by using contraction mapping principle. We refer the reader to [21,22,23,24] for some relevant results.

    Different from all the literature above listed, Hu et al. [25,26] researched the time-inconsistent stochastic linear quadratic (LQ, for short) control problems. They introduced the concept of equilibrium control by the local comparison made between open-loop controls. Using variational method, they derived the existence and uniqueness for equilibrium control, through a flow of forward-backward stochastic differential equations. In particular, they found an explicit equilibrium control and proved its uniqueness when the state is one dimensional and the coefficients in the problem are all deterministic. Some recent researched devoted to the open-loop equilibrium control can be found in [27,28,29] and the references therein.

    The necessary and sufficient conditions are the essential characterization of mathematical problems including optimal control problems. The classical LQ optimal control problems are the equivalent relationship between control problem, two-point boundary value problem, and Riccati equation. In particular, we would like to mention the work of Peng et al. in a very recent paper [30], they studied the equivalent relationship between equilibrium control, two-point boundary problem, and Riccati equation for the time-inconsistent deterministic LQ control. Additionally, He and Jiang [31] formally acquired a necessary and sufficient condition by a method of extended HJB equations on the equilibrium strategies for time-inconsistent problems in continuous time.

    Inspired by the above works, using the method of duality analysis, we study the necessary and sufficient conditions of optimal control problems with time inconsistency in the framework of open-loop equilibrium control in this paper. Furthermore, under the assumption of the solvability for the Riccati type equation, we investigate the existence of explicit equilibrium control by proving the solvability of two-point boundary value problem.

    The remainder of this paper is organized as follows. In the second section, we formulate the mathematical model for a general class of optimal control problem with time inconsistency and introduce the definition of equilibrium control in the sense of open-loop. Section 3 is devoted to present the main results for a general class of optimal control problem with time inconsistency. In Section 4, we consider LQ optimal control problem with time inconsistency.

    Starting with an optimal control problem with time inconsistency. Let T>0 be the end of a finite time horizon and URm.

    For any initial pair (t,x)[0,T]×Rn, the following controlled system can be considered.

    {˙Y(s)=h(s,Y(s),u(s)),s[t,T],Y(t)=x, (2.1)

    where h:[0,T]×Rn×URn is a given map, u(), a function valued in U, represents control, and Y()Rn represents the control trajectory. The control processes which are essentially bounded with respect to t, i.e.,

    U[0,T]=L([0,T];U).

    Subsequently, the following cost functional can be introduced.

    K(t,x;u())=Ttϕ(t,x;s,Y(s),u(s))ds+ψ(t,x;Y(T)). (2.2)

    for some given maps ϕ:[0,T]×Rn×[0,T]×Rn×UR and ψ:[0,T]×Rn×RnR. Under some mild conditions, for any (t,x)[0,T]×Rn and u()U[t,T], the state equation (2.1) admits a unique solution Y()Yut,x(), and the cost functional K(t,x;u()) is well-defined. Thus, the following problem can be introduced.

    Problem (TIP). For (t,x)[0,T]×Rn, find a control ˉu()U[t,T] such that

    K(t,x;ˉu())=infu()U[t,T]K(t,x;u()). (2.3)

    Problem (TIP) is an optimal control problem with time inconsistency. With the time inconsistency, the notion "optimality" needs to be defined in an appropriate way. We adopt the concept of equilibrium solution within the framework of open-loop in this paper.

    Definition2.1. [25] Let ˉu()U[0,T] be a given control and ˉY() be the control trajectory corresponding to ˉu(). The control ˉu() is called an equilibrium control if the following inequality holds

    limε0K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε0,(t,v)[0,T]×U, (2.4)

    where

    uε,t,v(s)={ˉu(s),0st,v,t<st+ε,ˉu(s),t+ε<sT. (2.5)

    In this paper, represents an Euclidean norm.

    Next, we make the following assumptions.

    [H] 1) The map h(s,y,u):[0,T]×Rn×URn be continuous, and h(s,y,u) be also continuous differential with respect to y. There exists a constant L>0 such that

    {h(s,y1,u1)h(s,y2,u2)∣≤L(y1y2|+|u1u2),h(s,0,u)∣≤L, (2.6)

    and

    {hy(s,y1,u)hy(s,y2,u)∣≤Ly1y2,hy(s,0,u)∣≤L, (2.7)

    for any s[0,T], y1,y2Rn, and u,u1,u2U.

    2) The maps ϕ(t,x;s,y,u):[0,T]×Rn×[0,T]×Rn×UR and ψ:[0,T]×Rn×RnR are continuous, and ϕ and ψ are also continuous differential with respect to y. There exists a constant L>0 such that

    {ϕy(t,x;s,y1,u)ϕy(t,x;s,y2,u)∣≤L|y1y2|,ϕy(t,x;s,0,u)∣≤L, (2.8)

    and

    {ψy(t,x;y1)ψy(t,x;y2)∣≤Ly1y2,ψy(t,x;0)∣≤L, (2.9)

    for any s[t,T], y1,y2Rn, and uU.

    In this section, we present a general necessary and sufficient condition of equilibrium control for the Problem (TIP).

    Let ˉu() is a given control and consider the perturbation control uε,t,v() defined by (2.5). Yε() and ˉY() be corresponding control processes to the control system (2.1) with uε,t,v() and ˉu(), respectively. We then introduce the following notations.

    {Φε(s,τ)=exp[sτ10hy(r,ˉY(r)+θ(Yε(r)ˉY(r)),ˉu(r))dθdr],Φ(s,τ)=exp[sτhy(r,ˉY(r),ˉu(r))dr], (3.1)

    for any 0τsT. And

    {φε(τ)=TτΦε(s,τ)10ϕy(t,ˉY(t);s,ˉY(s)+θ(Yε(s)ˉY(s)),ˉu(s))dθds+Φε(T,τ)10ψy(t,ˉY(t);ˉY(T)+θ(Yε(T)ˉY(T)))dθ,φ(τ)=TτΦ(s,τ)ϕy(t,ˉY(t);s,ˉY(s),ˉu(s))ds+Φ(T,τ)ψy(t,ˉY(t);ˉY(T)), (3.2)

    for any 0τT.

    Proposition3.1. Let [H] hold. Then

    1) Φε(,) and Φ(,) are uniform bounded in C([0,T]×[0,T]).

    2) Φε(,)Φ(,) a.e. in C([0,T]×[0,T]) as ε0.

    Proof. Let Yε() and ˉY() be the corresponding solutions to the control system (2.1) with uε,t,v() and ˉu(), respectively, where uε,t,v() is defined by (2.5) and ˉu() is a given control. Define

    {ˉY(s)=x+sth(τ,ˉY(τ),ˉu(τ))dτ,s[t,T],Yε(s)=x+sth(τ,Yε(τ),uε,t,v(τ))dτ,s[t,T].

    Then,

    ˉY(s)≤∣x+sth(τ,ˉY(τ),u(τ))dτ≤∣x+sth(τ,ˉY(τ),ˉu(τ))h(τ,0,ˉu(τ))ds+sth(τ,0,ˉu(τ))dτ≤∣x+LstˉY(τ)ds+stLds≤∣x+LT+LstˉY(τ)dτ.

    By Gronwall's inequality, one has

    ˉY(τ)∣≤(x+LT)eLT. (3.3)

    Similarly, we obtain that

    Yε(τ)∣≤(x+LT)eLT. (3.4)

    Moreover,

    Yε(s)ˉY(s)∣≤sth(τ,Yε(τ),uε,t,v(τ))h(τ,ˉY(τ),ˉu(τ))dτstL(Yε(τ)ˉY(τ)|+|uε,t,v(τ)ˉu(τ))dτ.

    Because

    stuε,t,v(τ)ˉu(τ)dτ={stvˉu(τ)dτ,s[t,t+ε],t+εtvˉu(τ)dτ,s(t+ε,T].

    Since u()L([t,T];U) for any t[0,T], there is a constant M>0 such that

    stuε,t,v(τ)ˉu(τ)dτt+εtvˉu(τ)dτ2Mεa.e.in[0,T]. (3.5)

    Gronwall's inequality yields

    Yε(s)ˉY(s)∣≤stesτLdrL(uε,t,v(τ)ˉu(τ))dτ2MLeLTεa.e.in[0,T].

    Passing to the limit in above as ε0 we obtain

    Yε()ˉY()a.e.in[0,T]. (3.6)

    Therefore, we have

    Φε(s,τ)=exp{sτ10{[hy(r,ˉY(r)+θ(Yε(r)ˉY(r)),ˉu(r))hy(r,ˉY(r),ˉu(r))]+[hy(r,ˉY(r),ˉu(r))hy(r,0,ˉu(r))]+hy(r,0,ˉu(r))}dθdr}.

    Combining (3.3) and (3.4), we then have

    Φε(s,τ)exp{sτ10hy(r,ˉY(r)+θ(Yε(r)ˉY(r)),ˉu(r))hy(r,ˉY(r),ˉu(r))+hy(r,ˉY(r),ˉu(r))hy(r,0,ˉu(r))+hy(r,0,ˉu(r))}dθdr}exp{sτ10{LθYε(r)ˉY(r)+LˉY(r)+L}dθdr}exp{sτ{LYε(r)ˉY(r)+LˉY(r)+L}dr}expsτ{2L(x+LT)eLT+L(x+LT)eLT+L}drexp[2L(x+LT)eLT+L(x+LT)eLT+L]T=exp[3L(x+LT)eLT+L]T.

    This is implies that Φε(,) is uniform bounded in C([0,T]×[0,T]). Similarly, we can easy prove Φ(,) is uniform bounded in C([0,T]×[0,T]). We thus completes the proof of (1).

    We now claim that Φε(,)Φ(,) a.e. in C([0,T]×[0,T]) as ε0. Since

    Φε(s,τ)Φ(s,τ)=exp{sτ10[hy(r,ˉY(r)+θ(Yε(r)ˉY(r)),ˉu(r))]dθdr}Φ(s,τ)=exp{sτ10[hy(r,ˉY(r)+θ(Yε(r)ˉY(r)),ˉu(r))hy(r,ˉY(r),ˉu(r))+hy(r,ˉY(r),ˉu(r))]dθdr}Φ(s,τ)=exp{sτ10[hy(r,ˉY(r)+θ(Yε(r)ˉY(r)),ˉu(r))hy(r,ˉY(r),ˉu(r))]dθdr}Φ(s,τ)Φ(s,τ)={exp{sτ10[hy(r,ˉY(r)+θ(Yε(r)ˉY(r)),ˉu(r))hy(r,ˉY(r),ˉu(r))]dθdr}Φ(s,s)}Φ(s,τ),

    which suggests that

    Φε(s,τ)Φ(s,τ)[exp(sτ10LθYε(r)ˉY(r)dθdr)Φ(s,s)]Φ(s,τ)[exp(sτLYε(r)ˉY(r)dr)1]Φ(s,τ).

    It follows from (3.6) and uniform boundedness of Φ(,) that

    Φε(s,τ)Φ(s,τ)∣→∣e01Φ(s,τ)∣=0asε0.

    This implies that (2) holds. We thus complete the proof.

    Proposition3.2. Let [H] hold. Then

    φε()φ()a.e. inC([0,T])asε0.

    Proof. By (3.2), we have

    φε(τ)φ(τ)=TτΦε(s,τ)10ϕy(t,ˉY(t);s,ˉY(s)+θ(Yε(s)ˉY(s)),ˉu(s))dθdsTτΦ(s,τ)ϕy(t,ˉY(t);s,ˉY(s),ˉu(s))ds+Φε(T,τ)10ψy(t,ˉY(t);ˉY(T)+θ(Yε(T)ˉY(T)))dθΦ(T,τ)ψy(t,ˉY(t);ˉY(T))=TτΦε(s,τ)10[ϕy(t,ˉY(t);s,ˉY(s)+θ(Yε(s)ˉY(s)),ˉu(s))ϕy(t,ˉY(t);s,ˉY(s),ˉu(s))]dθds+Tτ[Φε(s,τ)Φ(s,τ)][ϕy(t,ˉY(t);s,ˉY(s),ˉu(s))ϕy(t,ˉY(t);s,0,ˉu(s))+ϕy(t,ˉY(t);s,0,ˉu(s))]ds+Φε(T,τ)10[ψy(t,ˉY(t);ˉY(T)+θ(Yε(T)ˉY(T)))ψy(t,ˉY(t);ˉY(T))]dθ+[Φε(T,τ)Φ(T,τ)][ψy(t,ˉY(t);ˉY(T))ψy(t,ˉY(t);0)+ψy(t,ˉY(t);0)],

    which implies that

    φε(τ)φ(τ)Tτ10Φε(s,τ)LθYε(s)ˉY(s)dθds+TτΦε(s,τ)Φ(s,τ)L(ˉY(s)+1)ds+10Φε(T,τ)LθYε(T)ˉY(T)dθ+Φε(T,τ)Φ(T,τ)L(ˉY(T)+1)TτΦε(s,τ)LYε(s)ˉY(s)ds+TτΦε(s,τ)Φ(s,τ)L(ˉY(s)+1)ds+Φε(T,τ)LYε(T)ˉY(T)+Φε(T,τ)Φ(T,τ)L(ˉY(T)+1).

    It follows from Proposition 3.1, (3.3) and (3.6) that

    φε(τ)φ(τ)∣→0a.e.asε0.

    This completes the proof.

    Theorem3.3. Let [H] hold. Then Problem(TIP) admits an equilibrium control by Definition 2.1 if and only if the following quasi-variational problem

    {ˉY(t)=y0+t0h(s,ˉY(s),ˉu(s))ds,ω(t)=TtΦ(s,t)ϕY(t,ˉY(t);s,ˉY(s),ˉu(s))ds+Φ(T,t)ψY(t,ˉY(t);ˉY(T)),ˉu(t)argminvUH(t,ˉY(t);t,ˉY(t),v,ω(t)), (3.7)

    have a solution in C([0,T];Rn)×C([0,T];Rn), where

    H(t,x;s,y,u,p)=p,h(s,x,u)+ϕ(t,x;s,y,u), (3.8)

    for any (t,x;s,y,u,p)[0,T]×Rn×[0,T]×Rn×U×Rn.

    Proof. Let ˉu() is a given control and uε,t,v() is defined by (2.5), Yε() and ˉY() are the corresponding control processes to the control system (2.1) with uε,t,v() and ˉu(), respectively. Then we have

    {ˉY(s)=y0+s0h(τ,ˉY(τ),ˉu(τ))dτ,s[0,T],Yε(s)=y0+s0h(τ,Yε(τ),uε,t,v(τ))dτ,s[0,T].

    Define

    Zε(s)=1ε[Yε(s)ˉY(s)],s[0,T]. (3.9)

    This implies that Zε(0)=0 and

    Zε(s)={0,s[0,t],1εstΦε(s,τ)[h(τ,Yε(τ),v)h(τ,Yε(τ),ˉu(τ))]dτ,s(t,t+ε],Φε(s,t+ε)Zε(t+ε),s(t+ε,T]. (3.10)

    Now, we evaluate limε0K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε. It follows from (2.2) and (2.5) that

    K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=1εt+εt[ϕ(t,ˉY(t);s,Yε(s),v)ϕ(t,ˉY(t);s,ˉY(s),ˉu(s))]ds+Tt+ε10ϕY(t,ˉY(t);s,ˉY(s)+θεZε(s),ˉu(s))dθ,Zε(s)ds+10ψY(t,ˉY(t);ˉY(T)+θεZε(T))dθ,Zε(T).

    Plugging (3.10) into the above, we obtain that

    K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=1εt+εt[ϕ(t,ˉY(t);s,Yε(s),v)ϕ(t,ˉY(t);s,Yε(s),ˉu(s))]ds+1εt+εtTt+εΦε(s,τ)10ϕY(t,ˉY(t);s,ˉY(s)+θεZε(s),ˉu(s))dθds,h(τ,Yε(τ),v)h(τ,Yε(τ),ˉu(τ))dτ+1εt+εtΦε(T,τ)10ψY(t,ˉY(t);ˉY(T)+θεZε(T))dθ,h(τ,Yε(τ),v)h(τ,Yε(τ),ˉu(τ))dτ.

    Combining (3.8), (3.6) and Proposition 3.2, we then have

    limε0K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=H(t,ˉY(t);t,ˉY(t),v,ω(t))H(t,ˉY(t);t,ˉY(t),ˉu(t),ω(t)),a.e.t[0,T]. (3.11)

    On the one hand, if ˉu() is an equilibrium control, then

    H(t,ˉY(t);t,ˉY(t),v,ω(t))H(t,ˉY(t);t,ˉY(t),ˉu(t),ω(t)),a.e.t[0,T].

    This implies that

    ˉu()argminvUH(t,ˉY(t);t,ˉY(t),v,ω(t)),t[0,T]. (3.12)

    This completes the proof of necessary.

    Conversely, assume (ˉY(),ω())C([0,T];Rn)×C([0,T];Rn) be the solution of (3.7). Let ˉu() given by (3.7), we claim ˉu() is an equilibrium control. We consider the perturbation control uε,t,v() given by (2.5). Similarly to the calculation of (3.11), we can obtain that

    limε01ε[K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())]=H(t,ˉY(t);t,ˉY(t),v,ω(t))H(t,ˉY(t);t,ˉY(t),ˉu(t),ω(t)),a.e.t[0,T]. (3.13)

    By (3.12), we have

    H(t,ˉY(t);t,ˉY(t),v,ω(t))H(t,ˉY(t);t,ˉY(t),ˉu(t),ω(t))0,t[0,T].

    Therefore,

    limε01ε[K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())]0.

    We thus complete the proof.

    As application, we study an LQ optimal control problem with time inconsistency in this section.

    We use the following notations is this section.

    U[0,T]=L2([0,T];Rm).
    D[0,T]={(t,s)[0,T]2|0tsT}.
    ΦA(s,t)=exp{stA(r)dr},t,s[0,T].

    Now, we make the following standard assumptions:

    (Q1) AL1([0,T];Rn×n), BL2([0,T];Rn×m).

    (Q2) MC(D[0,T];Sm) is symmetry and positive definite.

    (Q3) LC(D[0,T];Sn), NC([0,T];Sn) are symmetry and positive semi-definite.

    (Q4) For 0tsT, ˙N(t) and Lt(t,s) are symmetry and positive semi-definite. Here

    Lt(t,s)=Lt(t,s).

    We consider the following LQ optimal control system from the situation (t,x)[0,T]×Rn.

    {˙Y(s)=A(s)Y(s)+B(s)u(s),s[t,T],Y(t)=x, (4.1)

    with the following cost functional

    K(t,x;u())=Tt[L(t,s)Y(s),Y(s)+M(t,s)u(s),u(s)]ds+N(t)Y(T),Y(T). (4.2)

    One could introduce the following control problem.

    Problem(TILQ). For (t,x)[0,T]×Rn. Find a control ˉu()U[0,T] such that

    K(t,x;ˉu())=infL2([0,T];Rm)K(t,x;u()). (4.3)

    Problem (TILQ) is an LQ optimal control problem with time inconsistency.

    Theorem4.1. Let (Q1)(Q4) hold. Then Problem (TILQ) admits an equilibrium control by Definition 2.1 if and only if the following two-point boundary value problem

    {ˉY(t)=ΦA(t,0)y0+t0ΦA(t,τ)B(τ)ˉu(τ)dτ,ϖ(t)=TtΦA(s,t)L(t,s)ˉY(s)ds+ΦA(T,t)N(t)ˉY(T),t[0,T], (4.4)

    have a solution in C([0,T];Rn)×C([0,T];Rn) and equilibrium control ˉu() is given by

    ˉu(t)=M1(t,t)B(t)ϖ(t),t[0,T]. (4.5)

    Proof. Let ˉY() and Yε() be the corresponding state trajectory to the control system (4.1) with ˉu() and uε,t,v(), respectively, where ˉu() is a given control and uε,t,v() is defined by (2.5). It is easy to prove that

    Yε()ˉY()inC([0,T];Rn)asε0. (4.6)

    Let

    Zε(s)=1ε[Yε(s)ˉY(s)],s[0,T]. (4.7)

    Then Zε(0)=0 and

    Zε(s)={0,s[0,t],1εstΦA(s,τ)B(τ)[vˉu(τ)]dτ,s(t,t+ε],1εt+εtΦA(s,τ)B(τ)[vˉu(τ)]dτ,s(t+ε,T]. (4.8)

    Now, we evaluate the variation of the cost functional. It follows from (4.2) and (2.5) that

    K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=1εt+εtM(t,s)[v+ˉu(s)],vˉu(s)ds+1εTtL(t,s)[Yε(s)+ˉY(s)],Yε(s)ˉY(s)ds+1εN(t)[Yε(T)+ˉY(T)],Yε(T)ˉY(T).

    Plugging (4.8) into the above, we can have

    K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=1εt+εtM(t,s)[v+ˉu(s)],vˉu(s)ds+1εt+εtB(τ)TτΦA(s,τ)L(t,s)[Yε(s)+ˉY(s)]ds,vˉu(τ)dτ+1εt+εtB(τ)ΦA(T,τ)N(t)[Yε(T)+ˉY(T)],vˉu(τ)dτ.

    It follows from (4.6) and (4.4) that

    limε0K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=2B(t)ϖ(t)+M(t,t)[v+ˉu(t)],vˉu(t). (4.9)

    This implies that

    limε0K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=M(t,t)[v+ˉu(t)+2M1(t,t)B(t)ϖ(t)],vˉu(t)0 (4.10)

    for any (t,v)[0,T]×Rm. We thus prove that ˉu(t)=M1(t,t)B(t)ϖ(t) is an equilibrium control by using the positive definiteness of the matrix-value function M.

    Conversely, suppose that Problem (TILQ) have an equilibrium control ˉu() and uε,t,v() is defined by (2.5). Let ˉY() and Yε() be the corresponding state trajectory to the control system (4.1) with ˉu() and uε,t,v(), respectively. Define

    ˜ϖ(t)=TtΦA(s,t)L(t,s)ˉY(s)ds+ΦA(T,t)N(t)ˉY(T). (4.11)

    Similarly to the computation of the (4.9), we have

    limε0K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε=M(t,t)[v+ˉu(t)]+2B(t)˜ϖ(t),vˉu(t),(t,v)[0,T]×Rm. (4.12)

    Define

    ˜K(t,ˉY(t);v)limε0K(t,ˉY(t);uε,t,v())K(t,ˉY(t);ˉu())ε,(t,v)[0,T]×Rm.

    It follows from (4.12) that ˜K(t,ˉY(t);v) is strictly convex in v. Therefore, the definition of an equilibrium control yields that ˜K(t,ˉY(t);v)0, which, together with (4.12), we obtain that ˜K(t,ˉY(t);v) admits a unique minimum point ˜v given by

    ˜v=M1(t,t)B(t)˜ϖ(t),t[0,T].

    Then the uniqueness of the minimum point ˜v yields that

    ˉu(t)=M1(t,t)B(t)˜ϖ(t),t[0,T]. (4.13)

    Combining (4.13) and (4.1), we then have

    {˙ˉY(t)=A(t)ˉY(t)B(t)M1(t,t)B(t)˜ϖ(t),ˉY(0)=y0. (4.14)

    Thus, the differential equation (4.14) admits a unique solution ˉY() given by

    ˉY(t)=ΦA(t,0)y0t0ΦA(t,τ)B(τ)M1(τ,τ)B(τ)˜ϖ(τ)dτ. (4.15)

    Plugging (4.15) into (4.11), we have

    ˜ϖ(t)=TtΦA(s,t)L(t,s)˜Y(s)ds+ΦA(T,t)N(t)˜Y(T). (4.16)

    Combining (4.15) and (4.16), we completes the proof.

    We now consider the solvability of the two-point boundary value problem (4.4). We introduce the following Riccati type equation:

    {˙Γ(t)+Γ(t)A(t)+AT(t)Γ(t)+ˉL(t,t)Γ(t)B(t)M1(t,t)B(t)Γ(t)=0,t[0,T],Γ(T)=N(T), (4.17)

    where

    ˉL(t,t)=L(t,t)ΦA(T,t)˙N(t)ˆΦ(T,t)TtΦA(τ,t)Lt(t,τ)ˆΦ(τ,t)dτ.

    Here

    ˆΦ(τ,t)=exp{τt[A(r)B(r)M1(r)B(r)Γ(r)]dr}, (4.18)

    for any 0tτT. Observe that ˆΦ(,) depends on the unknown term Γ(). Moreover, the Riccati type equation (4.17) is different from that of [30]. Therefore the results in [30] cannot be applied directly to (4.17) for its well posedness.

    It is also worth pointing out that (4.17) does not have a symmetric structure. Thus Γ() is not expected to be symmetric.

    Theorem4.2. Let Assumptions (Q1)(Q4) hold. Suppose that the Riccati type equation (4.17) admits a unique solution Γ()C([0,T];Rn×n). Then the two-point boundary value problem (4.4) admits a solution (ˉY(),ϖ())C([0,T];Rn)×C([0,T];Rn). Furthermore, for any t[0,T], we have

    ˉY(t)=ˆΦ(t,0)y0, (4.19)
    ϖ(t)=Γ(t)ˉY(t), (4.20)
    ˉu(t)=M1(t,t)B(t)Γ(t)ˉY(t), (4.21)

    where ˆΦ(t,0) is as introduced in (4.18).

    Proof. Suppose that Γ()C([0,T];Rn×n) is a solution of (4.17) and define (ˉY(),ϖ(),ˉu()) as in (4.19)–(4.21). We are going to show that ˉY() and ϖ() satisfy the two-point boundary value problem (4.4). First, observe that ˉY() satisfies the following ODE,

    {˙ˉY(t)=[A(t)B(t)M1(t,t)B(t)Γ(t)]ˉY(t)=A(t)ˉY(t)+B(t)ˉu(t),t[0,T],ˉY(0)=y0. (4.22)

    On the other hand, differentiating both sides of (4.20) with respect to t, we obtain that

    ˙ϖ(t)=˙Γ(t)ˉY(t)+Γ(t)˙ˉY(t)=[Γ(t)A(t)+AT(t)Γ(t)+ˉL(t,t)Γ(t)B(t)M1(t,t)B(t)Γ(t)]ˉY(t)+Γ(t)[A(t)B(t)M1(t,t)B(t)Γ(t)]ˉY(t)=A(t)Γ(t)ˉY(t)ˉL(t,t)ˉY(t),

    which implies that

    ϖ(t)=ΦA(T,t)N(T)ˉY(T)+TtΦA(τ,t)ˉL(τ,τ)ˉY(τ)dτ=ΦA(T,t)N(T)ˉY(T)TtΦA(τ,t)ΦA(T,τ)˙N(τ)ˆΦ(T,τ)ˉY(τ)dτ+TtΦA(τ,t)[L(τ,τ)TτΦA(s,τ)Lt(τ,s)ˆΦ(s,τ)ds]ˉY(τ)dτ=ΦA(T,t)[N(T)Tt˙N(τ)dτ]ˉY(T)+TtΦA(τ,t)L(τ,τ)ˉY(τ)dτTtTτΦA(τ,t)ΦA(s,τ)Lt(τ,s)ˆΦ(s,τ)ˉY(τ)dsdτ. (4.23)

    Observe that

    TtΦA(τ,t)L(τ,τ)ˉY(τ)dτTtTτΦA(τ,t)ΦA(s,τ)Lt(τ,s)ˆΦ(s,τ)ˉY(τ)dsdτ=TtΦA(τ,t)L(τ,τ)ˉY(τ)dτTtTτΦA(s,t)Lt(τ,s)ˉY(s)dsdτ=TtΦA(s,t)L(s,s)ˉY(s)dsTtΦA(s,t)stLt(τ,s)dτˉY(s)ds=TtΦA(s,t)[L(s,s)stLt(τ,s)dτ]ˉY(s)ds=TtΦA(s,t)L(t,s)ˉY(s)ds.

    Invoking this into (4.23), we obtain that

    ϖ(t)=ΦA(T,t)N(t)ˉY(T)+TtΦA(s,t)L(t,s)ˉY(s)ds. (4.24)

    It follows from (4.22) and (4.24) that the two-point boundary value problem (4.4) admits a solution in C([0,T];Rn)×C([0,T];Rn).This completes the proof.

    Combining Theorem 4.1 with Theorem 4.2, we obtain the following result.

    Corollary4.3. Let Assumptions (Q1)(Q4) hold. Suppose that the Riccati type equation (4.17) admits a unique solution Γ()C([0,T];Rn×n). Then Problem (TILQ) has an equilibrium control that can be represented by the state feedback form

    ˉu(t)=M1(t,t)B(t)Γ(t)ˉY(t),t[0,T].

    The author gratefully acknowledges to the anonymous reviewers and editor for their careful review and valuable comments and supported by National Natural Science Foundation of China grant No. 12061021.

    The author declares there is no conflict of interest.



    [1] R. Bellman, Dynamic Programming, Princeton University Press, New Jersey, 1957.
    [2] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mischenko, The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, 1962.
    [3] D. Hume, A Treatise of Human Nature, Oxford University Press, New York, 1978.
    [4] A. Smith, The Theory of Moral Sentiments, Oxford University Press, New York, 1976.
    [5] V. Pareto, Manuel Dˊeconomie Politique, Girard and Brieve, Paris, 1909.
    [6] P. Samuelson, A note on measurement of utility, Rev. Econ. Stud., 4 (1937), 155–161. https://doi.org/10.2307/2967612 doi: 10.2307/2967612
    [7] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Rev. Econ. Stud., 23 (1955), 165–180. https://doi.org/10.2307/2295722 doi: 10.2307/2295722
    [8] H. Thaler, Asymmetric games and the endowment effect, Behav. Brain Sci., 7 (1984), 117. https://doi.org/10.1017/S0140525X00026492 doi: 10.1017/S0140525X00026492
    [9] F. Kydland, E. Prescott, Rules rather than discretion: The inconsistency of optimal plans, J. Polit. Econ., 85 (1977), 473–491. https://doi.org/10.1086/260580 doi: 10.1086/260580
    [10] D. Kahneman, A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263–291. https://doi.org/10.2307/1914185 doi: 10.2307/1914185
    [11] I. Ekeland, A. Lazrak, Being serious about non-commitment: Subgame perfect equilibrium in continuous time, preprint, arXiv: math/0604264.
    [12] T. Bj¨ork, A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, 2010. Available from: https://ssrn.com/abstract=1694759.
    [13] T. Bj¨ork, M. Khapko, A. Murgoci, On time inconsistent stochastic control in continuous time, Financ. Stoch., 21 (2017), 331–360. https://doi.org/10.1007/s00780-017-0327-5 doi: 10.1007/s00780-017-0327-5
    [14] T. Bj¨ork, A. Murgoci, A general theory of Markovian time inconsistent stochastic control in discrete time, Financ. Stoch., 18 (2014), 545–592. https://doi.org/10.1007/s00780-014-0234-y doi: 10.1007/s00780-014-0234-y
    [15] T. Bj¨ork, M. Khapko, A. Murgoci, Time inconsistent stochastic control in continuous time: Theory and examples, preprint, arXiv: 1612.03650.
    [16] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations-time consistent solutions, Trans. Am. Math. Soc., 369 (2017), 5467–5523. https://doi.org/10.1090/tran/6502 doi: 10.1090/tran/6502
    [17] J. Yong, Time-inconsistent optimal control problem and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271–329. https://doi.org/10.3934/mcrf.2012.2.271 doi: 10.3934/mcrf.2012.2.271
    [18] J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Relat. Fields, 1 (2011), 83–118. https://doi.org/10.3934/mcrf.2011.1.83 doi: 10.3934/mcrf.2011.1.83
    [19] J. Yong, Deterministic time-inconsistent optimal control problem-an essentially cooperative approach, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 1–30. https://doi.org/10.1007/s10255-012-0120-3 doi: 10.1007/s10255-012-0120-3
    [20] Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Appl. Math. Optim., 84 (2021), 567–588. https://doi.org/10.1007/s00245-020-09654-7 doi: 10.1007/s00245-020-09654-7
    [21] G. Zhang, Q. Zhu, Event-triggered optimized control for nonlinear delayed stochastic systems, IEEE Trans. Circuits Syst. I Regul. Pap., 68 (2021), 3808–3821. https://doi.org/10.1007/s00245-020-09654-7 doi: 10.1007/s00245-020-09654-7
    [22] Q. Wei, J. Yong, Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM J. Control Optim., 55 (2019), 4156–4201. https://doi.org/10.1137/16M1079415 doi: 10.1137/16M1079415
    [23] T. Wang, Z. Jin, J. Wei, Mean-variance portfolio selection under a non-Markovian regime-switching model: Time-consistent solutions, SIAM J. Control Optim., 57 (2019), 3240–3271. https://doi.org/10.1137/18M1186423 doi: 10.1137/18M1186423
    [24] F. Dou, Q. Lv, Time-inconsistent linear quadratic optimal control problems for stochastic evolution equations, SIAM J. Control Optim., 58 (2020), 485–509. https://doi.org/10.1137/19M1250339 doi: 10.1137/19M1250339
    [25] Y. Hu, H. Jin, X. Y. Zhou, Time inconsistent stochastic linear quadratic control, SIAM J. Control Optim., 50 (2012), 1548–1572. https://doi.org/10.1137/110853960 doi: 10.1137/110853960
    [26] Y. Hu, H. Jin, X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261–1279. https://doi.org/10.1137/15M1019040 doi: 10.1137/15M1019040
    [27] I. Alia, Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems, Math. Control Relat. Fields, 11 (2021), 1–44. https://doi.org/10.3934/mcrf.2021053 doi: 10.3934/mcrf.2021053
    [28] I. Alia, Time-inconsistent stochastic optimal control problems: A backward stochastic partial differential equations approach, Math. Control Relat. Fields, 10 (2020), 785–826. https://doi.org/10.3934/mcrf.2020020 doi: 10.3934/mcrf.2020020
    [29] I. Alia, A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Math. Control Relat. Fields, 9 (2019), 541–570. https://doi.org/10.3934/mcrf.2019025 doi: 10.3934/mcrf.2019025
    [30] H. Cai, D. Chen, Y. Peng, W. Wei, On the time-inconsistent deterministic linear-quadratic control, SIAM J. Control Optim., 60 (2022), 968–991. https://doi.org/10.1137/21M1419611 doi: 10.1137/21M1419611
    [31] X. He, L. Jiang, On the equilibrium strategies for time-inconsistent problem in continuous time, SIAM J. Control Optim., 59 (2019), 3860–3886. https://doi.org/10.1137/20M1382106 doi: 10.1137/20M1382106
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