Wellposedness of a class of twopoint boundary value problems associated with ordinary differential equations
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Abstract
This paper introduces the regular decoupling field to study the existence and uniqueness of solutions of twopoint boundary value problems for a class of ordinary differential equations which can be derived from the maximum principle in optimal control theory. The monotonicity conditions used to guarantee the existence and uniqueness of such equations are initially a special case of the regular decoupling field method. More generally, in case of the homogeneous equations, this paper generalizes the application scope of the monotonicity conditions method by using the linear transformation method. In addition, the linear transformation method can be used to handle the situation where the monotonicity conditions and regular decoupling field method cannot be directly applied. These two methods overall develop the wellposedness theory of twopoint boundary value problems which has potential applications in optimal control and partial differential equation theory.
Keywords
Ordinary differential equations Twopoint boundary value problems Regular decoupling field Monotonicity conditions Linear transformation Optimal control theory1 Introduction
In this paper, only a onedimensional case is considered for simplicity, and the multidimensional cases are dealt similarly. The purpose is to find a pair of \((X_{t},Y_{t}) \in C[0,T]\), for arbitrary \(T>0\), to satisfy ODEs (1.1), which is called the wellposedness study. These are twopoint boundary value problems for a class of ODEs. The Hamilton system derived from the Pontryagin maximum principle, which is a milestone of the optimal control theory, belongs to this class. This kind of ODEs can also be related to one kind of partial differential equations (PDEs) (see, for example, [1]). Solving such equations is of great significance in the field of optimal control. Only the linear case is discussed in this paper. Actually, the optimal state system, which comes from the classical linear quadratic (LQ) optimal control problem combined with the adjoint equations, belongs to this kind of ODEs. Therefore, the wellposedness of twopoint boundary value problems for such ODEs on arbitrary time duration has very meaningful application background and practical significance. As can be observed in ODEs (1.1), the equations of \(X_{t}\) and \(Y_{t}\) both have \((X_{t},Y_{t})\) as their components, which makes two equations fully coupled together. It is impossible to solve each equation individually, then many methods adapted to ODEs with one unknown variable are no longer feasible (see, for example, [2, 3]).
Such kind of equations becomes a stochastic Hamilton system when taking random noise into consideration, which also can be called the forward–backward stochastic differential equations (FBSDEs). The wellposedness of FBSDEs is also hard to get, and it has widely practical applications in the field of stochastic optimal control as well as financial mathematics. On the wellposedness of FBSDEs on arbitrary time duration, Hu and Peng [4] and Peng and Wu [5] introduce the method of continuation by proposing the monotonicity conditions (see [6]). The existence and uniqueness of solutions are obtained by this method. By using the method of continuation, Wu [7] weakens the monotonicity conditions and obtains the existence and uniqueness of the solutions to twopoint boundary value problems for ODEs (1.1) and also the corresponding comparison theorem. However, the monotonicity conditions have a strict restriction on coefficients. The method of continuation can only be used in some certain situations of ODEs (1.1). In case of FBSDEs, Ma et al. [8] introduce the unified approach which leads to the wellposedness of FBSDEs by means of regular decoupling field. The unified approach generalizes the work of solvability of FBSDEs, which makes monotonicity conditions a special case of the unified approach.
In this paper, the regular decoupling field method and the linear transformation method are introduced to study the existence and uniqueness of solutions of twopoint boundary value problems for ODEs (1.1). The linear transformation method generalizes the application scope of monotonicity conditions. And these two methods develop the work of solvability of ODEs (1.1), which makes contribution to the optimal control theory.
After giving the preliminaries and assumptions in Section 2, the rest of the paper is organized as follows. In Section 3, for the twopoint boundary value problems of ODEs (1.1), the regular decoupling field is introduced to guarantee the wellposedness of such ODEs. Moreover, in Section 4, for some cases that cannot be applied with the regular decoupling field directly, linear transformation method is introduced to weaken the coefficients restriction of the monotonicity conditions. These two methods develop the wellposedness theory of twopoint boundary value problems for ODEs, and it is feasible to be applied to the research of the control theory and PDEs. At last, we conclude the main results of this paper and give the future research direction in Section 5.
2 Preliminaries and assumptions
We first assume the following.
Assumption 2.1
 (i)
Homogeneous coefficients \(a_{t}\), \(b_{t} \), \(c_{t}\), \(d_{t}\) are uniformly boundary on \([0,T]\).
 (ii)Inhomogeneous coefficients \(f_{t}\) and \(g_{t}\) hold:$$\int_{0}^{T} {f_{t}}^{2} \,dt < \infty,\qquad \int_{0}^{T} {g_{t}}^{2} \,dt < \infty. $$
From Zhang’s lemma [9], it is easy to get the wellposedness of ODEs (1.1) on small duration.
Theorem 2.2
\(\exists\delta> 0\), whenever\(T \leq\delta\), if the terminal function is uniformly Lipschitz continuous in its spatial variable, ODEs (1.1) have a unique solution on\([0,T]\).
As can be seen, when \(T<\delta\), the terminal function of ODEs (1.1) is a linear function for \(X_{T}\), and \(H\in R\). Then the wellposedness is obvious according to Theorem 2.2. We introduce the decoupling field to study the wellposedness of ODEs (1.1) when T is arbitrary duration.
Definition 2.3
From the result of Ma et al. [8], we have the following.
3 Regularity of decoupling field
Equation (3.3) is called the “characteristic equation” of (1.1). According to the analysis in the last section, studying the wellposedness of ODEs (1.1) is essentially finding conditions that ensure the solution of equation (3.3) û bounded on \([0,T]\).
As can be seen from (3.4), equation (3.3) is a Riccati equation. The Riccati equation can be solved when knowing a specific solution of it. Otherwise, we should introduce the following comparison theorem to get the boundedness of solution of equation (3.3).
Lemma 3.1
 (i)
Equations (3.6) always have bounded solution\(\overline {y}_{t}\), \(\underline{y}_{t} \), \(t\in[0,T]\).
 (ii)
For every\(t\in[0,T]\), the function\(y\mapsto F(t,y)\) (\(\underline{F}(y) \)or\(\overline{F}(y)\)) is uniformly Lipschitz continuous on\(y\in[\underline{y}_{t},\overline{y}_{t}]\), where the Lipschitz constant isL.
 (iii)
For any\(t\in[0,T]\), \(C^{i}\geq\int_{t}^{T} e^{\int_{s}^{T} \alpha _{r} \,dr}g_{s}^{i}\,ds\)always holds, whereαmatches\(\alpha\leq L\).
Then equation (3.5) has a unique solutionymatching\(\underline {y}\leq y \leq\overline{y}\).
Remark 3.2
Equations (3.6) are called the upper/lower boundary equations of ODE (3.5). The classical sufficient conditions for situation(iv) are: \(C^{i}\geq\int_{0}^{T} e^{L(Tt)}(g^{i}_{t})^{+}dt\). Particularly, that is satisfied if \(C^{i}=0\) and \(g^{i}\leq0\).
Case 1: ConstantCoefficient
Theorem 3.3
 (i)
\(F(H)\geq0\), and F has a zero point in\([H,\infty]\).
 (ii)
\(F(H)\leq0\), and F has a zero point in\([\infty,H]\).
 (iii)
\(b=0\).
Proof
(ii) can be proved similarly.
Theorem 3.3(iii) is easy to check, but (i) and (ii) are not directly connected with the coefficients. Next, we give some equivalence conditions.

\(b<0\), \(F(H)\geq0\),

\(b>0\), \(F(H)\geq0\), \((a+d)^{2}4bc\geq0\), \(H\leq\frac{a+d}{2b}\).

\(b>0\), \(F(H)\leq0\),

\(b<0\), \(F(H)\leq0\), \((a+d)^{2}4bc\geq0\), \(H\geq\frac{a+d}{2b}\).
Situations matching Theorem 3.3
Case  b  F(H)  Assumptions 

1  b<0  F(H)≥0  
2  b>0  F(H)≥0  \((a+d)^{2}4bc\geq0\), \(H\leq\frac{a+d}{2b}\) 
3  b>0  F(H)≤0  
4  b<0  F(H)≤0  \((a+d)^{2}4bc\geq0\), \(H\geq\frac{a+d}{2b}\) 
Next, we focus on the connections between Theorem 3.3 and the monotonicity conditions. According to Peng and Wu [5] and Wu [7], matching one of the following conditions, we can also get the wellposedness of ODEs (1.1) on \([0,T]\).
Lemma 3.4
(Monotonicity conditions)
 (i)where\(\beta_{1}\)and\(\beta_{2}\)are nonnegative constants. When\(\beta_{1} > 0\), \(H > 0\), then\(\beta_{2} \geq0\); when\(\beta_{2} > 0\), then\(\beta_{1} \geq0\), \(H \geq0\).$\left(\begin{array}{cc}x& y\end{array}\right)\left(\begin{array}{cc}c& d\\ a& b\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\le {\beta}_{1}x{}^{2}{\beta}_{2}y{}^{2},$(3.10)
 (ii)where\(\beta_{1}\)and\(\beta_{2}\)are nonnegative constants. When\(\beta_{1} > 0\), \(H < 0\), then\(\beta_{2} \geq0\); when\(\beta_{2} > 0\), then\(\beta_{1} \geq0\), \(H \leq0\).$\left(\begin{array}{cc}x& y\end{array}\right)\left(\begin{array}{cc}c& d\\ a& b\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\ge {\beta}_{1}x{}^{2}+{\beta}_{2}y{}^{2},$(3.11)
According to Vieta’s theorem, Lemma 3.4 is equivalent to the following lemma.
Lemma 3.5
(Equivalence conditions)
According to Theorem 3.3, the analysis above means \(\hat{u}(t)\in [0,\lambda]\). Thus, Lemma 3.5(i) totally falls into the framework of Theorem 3.3.
Similarly, Lemma 3.5(ii) can be derived from case 2, case 3 of Table 1 and the case \(\hat{u}(t)\in[\lambda,0]\). It is concluded that the monotonicity conditions can be derived from Theorem 3.3, which means the monotonicity conditions are a special case of the regular decoupling field method.
An example is given where the coefficients match the assumption of Theorem 3.3. It is noted that this example does not match the monotonicity conditions.
Example 1
According to Lemma 3.4, the signs of H and b should be different to match the monotonicity conditions. Thus, the wellposedness of equations (3.14) cannot be proved by using the framework of monotonicity conditions.
Then ODEs (3.14) match Theorem 3.3(ii) (case 3 of Table 1). It is obvious that ODEs (3.14) have a unique solution thanks to the analysis above.
Case 2: Functional coefficients
In this part, we consider the case where coefficients of ODEs (1.1) are functions defined on \([0,T]\). In this case, \(F(s,\hat{u}(s))\) takes the form (3.4), and also Assumption 2.1 holds. It is noted that \(\overline {F}(t,\hat{u}(t))\)/\(\underline{F}(t,\hat{u}(t))\) defined in (3.7) is regarded as an upper/lower bound of \(F(s,\hat{u}(s))\). In analogy to Theorem 3.3, we have the following result.
Theorem 3.6
 (i)
\(\underline{F}(t,H)\geq0\), and\(\overline{F}(t,H)\)has a zero point in\([H,\infty]\).
 (ii)
\(\underline{F}(t,H)\leq0\), and\(\overline{F}(t,H)\)has a zero point in\([\infty,H]\).
 (iii)
\(b_{s}=0\).
Proof
(ii) can be proved similarly, and \(\hat{u}(t)\in[\lambda,H]\).
According to Theorem 3.3(iii), the analytic solution of the upper/lower bound equation can be derived, which means equation (3.4) has a bounded solution on \([0,T]\). □
From two cases above, it is concluded that the regular decoupling field can be used to prove the wellposedness of twopoint boundary value problems for ODEs (1.1), especially when ODEs (1.1) cannot match the monotonicity conditions. In the next section, the linear transformation method is used to generalize the framework of the monotonicity conditions.
4 The linear transformation method
Lemma 4.1
 (i)where\(\beta_{1}\)and\(\beta_{2}\)are nonnegative constants. Also, \(\beta _{1}\), \(\beta_{2}\), \(C_{2}\), andHcannot be 0 at the same time.$\begin{array}{rl}& \left(\begin{array}{cc}x& y\end{array}\right)\left(\begin{array}{cc}c& d\\ a& b\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\le {\beta}_{1}x{}^{2}{\beta}_{2}y{}^{2},\\ & {C}_{2}\le 0,\phantom{\rule{2em}{0ex}}H\ge 0,\end{array}$(4.5)
 (ii)where\(\beta_{1}\)and\(\beta_{2}\)are nonnegative constants. Also, \(\beta _{1}\), \(\beta_{2}\), \(C_{2}\), andHcannot be 0 at the same time.$\begin{array}{rl}& \left(\begin{array}{cc}x& y\end{array}\right)\left(\begin{array}{cc}c& d\\ a& b\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\ge {\beta}_{1}x{}^{2}+{\beta}_{2}y{}^{2},\\ & {C}_{2}\ge 0,\phantom{\rule{2em}{0ex}}H\le 0,\end{array}$(4.6)
Lemma 4.2
 (i)
\(\tilde{b}_{2}\leq0 \), \(C_{2}\leq0\), \(\tilde{H}\geq0\), \((mn)^{2}(b_{1}f_{2})^{2}+4f(m)f(n)<0\),
 (ii)
\(\tilde{b}_{2}\geq0 \), \(C_{2}\geq0\), \(\tilde{H}\leq0\), \((mn)^{2}(b_{1}f_{2})^{2}+4f(m)f(n)<0\),
As Lemma 4.2 is not convenient to check, then derivation is as follows.
In summary, we have the following result.
Theorem 4.3
In the following example, where the monotonicity conditions and regular decoupling field methods cannot be directly applied, the wellposedness of ODEs can be obtained by using the linear transformation method discussed in this section.
Example 2
It is easy to check that equations (4.11) match Lemma 4.1(ii) which means (4.11) has a unique solution.
5 Conclusion
In this paper, we introduce two methods to solve the twopoint boundary value problems of ODEs (1.1). The first method is the regular decoupling field which generates from the unified approach for FBSDEs. But for ODEs, the regular decoupling field method has a direct criterion which makes it easy to apply. Moreover, in this paper, it can be proved that the monotonicity conditions are a special case of the regular decoupling field method. The second method we introduce is the linear transformation method. It can be applied to cases where the monotonicity conditions and the regular decoupling field method can not. We also give examples in this paper to illustrate how these two methods develop the theory of the twopoint boundary value problems for ODEs which has meaningful applications in optimal control and PDEs theory. In addition, the linear transformation method can also be generalized into stochastic cases. This provides another way to study the wellposedness of FBSDEs, which is our future research direction and has some potential applications.
Notes
Acknowledgements
This work was supported by the Natural Science Foundation of China (61573217), the National Highlevel Personnel of Special Support Program and the Chang Jiang Scholar Program of Chinese Education Ministry.
Authors’ contributions
All authors have contributed equally in this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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