 Research
 Open Access
An optimal class of nondegenerate potentials for secondorder ordinary differential equations
 Kaiming Shen^{1} and
 Meirong Zhang^{1}Email author
 Received: 19 March 2015
 Accepted: 6 October 2015
 Published: 16 October 2015
Abstract
By considering the Dirichlet boundary condition \(x(0)=x(1)=0\), we say that \(q\in L^{1}[0,1]\) is a nondegenerate potential if the ordinary differential equation \(x''+q(t) x=0\) has only the trivial solution \(x(t)\equiv0\) which verifies the boundary condition. Starting with a nondegenerate positive constant potential B, in this paper, we will apply the Pontryagin maximum principle (PMP) in optimal control theory to find the optimal bound \(r=r(A,B)\) for any \(A\in[\infty,B)\) such that any potential \(q\in L^{1}[0,1]\) satisfying \(A\le q \le B\) and \(\int_{[0,1]} q(t)\,dt >r(A,B)\) is necessarily nondegenerate. Such a nondegeneracy problem can be considered as the dual problem in a series of papers by Li et al.
Keywords
 nondegenerate potential
 eigenvalue
 optimal control
 Pontryagin maximum principle
 boundary value problem
1 Introduction
Let us introduce the notion of nondegenerate potentials by considering the Dirichlet boundary condition.
Definition 1
Lemma 2
For \(q\in L^{1}[0,1]\), problem (1)(2) is nondegenerate if and only if \(\lambda_{m}(q) \ne0\) for all \(m\in \mathbf{N}\).
In [17], the second author of this paper has used some Sobolev inequalities to characterize another important class of nonconstant nondegenerate potentials for problem (1)(2) and its pLaplacian counterpart. This class of nondegenerate potentials is used in [9, 10]. It is worth mentioning that papers [18–20] on the Lyapunovtype inequalities are also related with nondegenerate potentials.
Theorem 3

For \(A\in(0,\mu_{n}]\), letbe the unique solution of the following equation:$$ r:=r(A,B)\in \bigl[\mu_{n},A+{n\pi(BA)}/{\sqrt{B}} \bigr]\subset [\mu_{n},B) $$(9)$$ {\sqrt{B}} {\cot\frac{\sqrt{B}}{2n}\frac{rA}{BA}}={\sqrt{A}} { \tan \frac{\sqrt{A}}{2n}\frac{Br}{BA}}. $$(10)

For \(A\in[\infty ,0]\), let$$ r(A,B):=n\pi\sqrt{B}=\pi\sqrt{B}[\sqrt{B}/\pi]. $$(11)
One may compare equation (10) with (5). The results of Theorem 3 are obtained mainly using the PMP. However, different from the arguments in [3, 4, 6, 7], we will extensively apply the eigenvalue theory for problem (3). In fact, the nondegeneracy of potentials is a consequence of the estimates (12) on eigenvalues which are also optimal in a certain sense. Moreover, in order to prove the existence of the optimal control potentials for the optimal control problems deduced from the nondegenerate potential problems, we find that the (strong) continuous dependence of solutions and eigenvalues of (3) in potentials (with weak topology) in [21] can simplify the arguments significantly. In these senses, the present paper has given some simpler approach to the nondegeneracy problem.
The paper is organized as follows. At first, we introduce the Pontryagin maximum principle in optimal control theory and establish the connection between nondegeneracy problems and optimal control problems. Secondly, Theorem 3 is proved by solving equations in the PMP. Finally, we briefly consider nondegenerate potentials of (1) with the Neumann boundary condition and point out that the class of potentials in Theorem 3 is also nondegenerate with respect to the Neumann problem. As seen from applications of nondegenerate potentials to nonlinear differential equations mentioned above, it can be expected that the new class of nondegenerate potentials in Theorem 3 can lead to interesting applications to semilinear and superlinear differential equations.
2 Control systems and the Pontryagin maximum principle

the statecontrol trajectory \((\mathbf{x},\mathbf{u})\) is characterized by a firstorder ordinary differential systemwith a known initial state$$ \dot{\mathbf{x}}=\mathbf{f}\bigl(\mathbf{x},\mathbf{u}(t),t\bigr), \quad t\in [t_{0},t_{f}], $$(13)where \(\mathbf{x}, \mathbf{x}_{0}\in \mathbf{R}^{n}\), \(\mathbf{u}\in \mathbf{R}^{m}\);$$ \mathbf{x}(t_{0})=\mathbf{x}_{0}, $$(14)

the final state is usually described bywhere \(\mathbf{g}: \mathbf{R}^{n} \to \mathbf{R}^{k}\) is a known function;$$ \mathbf{g}\bigl(\mathbf{x}(t_{f})\bigr)=0, $$(15)

the set of admissible controls is described bywhere \(U_{m}\subset \mathbf{R}^{m}\) is a known domain; and$$\begin{aligned} \mathbf{U}_{[t_{0},t_{f}]} := & \bigl\{ \mathbf{u}(t): \mathbf {u}(\cdot) \mbox{ is a piecewise continuous function on }[t_{0},t_{f}] \\ &{}\mbox{such that }\mathbf{u}(t)\in U_{m}\mbox{ for all }t \in[t_{0},t_{f}] \mbox{ and} \\ &{}\mbox{problem (13)(14)(15) has solutions } \mathbf{x}(t) \bigr\} , \end{aligned}$$(16)

the cost functional is a functional of \(\mathbf{u}(\cdot)\in \mathbf{U}_{[t_{0},t_{f}]}\) taking the following form:where \(L(\mathbf{x},\mathbf{u},t)\) is a known function.$$ J\bigl[\mathbf{u}(\cdot)\bigr]=\int_{t_{0}}^{t_{f}}L \bigl(\mathbf{x}(t),\mathbf{u}(t),t\bigr)\,dt, $$(17)
The optimal control problem is to find an admissible control \(\mathbf {u}(\cdot)\in\mathbf{U}_{[t_{0},t_{f}]}\) that maximizes the cost functional \(J[\mathbf{u}(\cdot)]\). Suppose that the optimal control problem is solvable and \((\mathbf {x}^{*},\mathbf{u}^{*})\) is the optimal statecontrol trajectory . Then \(\mathbf{u}^{*}\) and \(\mathbf{x}^{*}\) can be characterized by the Pontryagin maximum principle [22].
Theorem 4
(Pontryagin maximum principle)

\(\mathbf{f}(\mathbf{x}, \mathbf{u},t)\), \(\mathbf {f}_{\mathbf{x}}(\mathbf{x},\mathbf{u},t)\), \(L(\mathbf{x},\mathbf {u},t)\), \(\mathbf{g}(\mathbf{x})\), \(\mathbf{g}_{\mathbf {x}}(\mathbf{x})\) are continuous; and

\(\mathbf{f}(\mathbf{x},\mathbf{u},t)\), \(\mathbf {f}_{\mathbf{x}}(\mathbf{x},\mathbf{u},t)\), \(L_{\mathbf{x}}(\mathbf {x},\mathbf{u},t)\) are bounded.
In the following we consider the case that \(A\in(\infty ,\mu_{n}]\) is finite.
Lemma 5
The optimal control problem (24)(27) associated with problem (1)(2) has an optimal control potential \(q^{*}\in \Omega _{A,B}\).
Proof
Lemma 6
Proof
Suppose that \(q\in \Omega _{A,B}\) satisfies (8). If problem (1)(2) has nontrivial solutions, one would have \(q\in\hat{\Omega }_{A,B}\) and \(J[q^{*}]=\sup_{\hat{q}\in\hat{\Omega }_{A,B}} J[\hat{q}] \ge J[q]\), a contradiction with (8).
Since \(q\equiv B\in \Omega _{A,B}\) is nondegenerate, one has \(q^{*}\prec B\) and \(r(A,B)< B\). On the other hand, as \(q\equiv\mu_{n}\in \Omega _{A,B}\) is degenerate, one has \(r(A,B)\ge J[\mu_{n}]=\mu_{n}\). These have given the bounds (28) for \(r(A,B)\). □
3 Construction of nondegenerate potentials for the Dirichlet problem
Lemma 7
The optimal control potential \(q^{*}\) in Lemma 5 must satisfy \(\lambda_{n}(q^{*})=0\). Consequently, as an eigenfunction, \(x_{1}^{*}(t)\) has precisely \((n+1)\) zeros in \([0,1]\), say \(0=t_{0}< t_{1} < \cdots< t_{n} =1\).
Proof
Since \(q^{*} \in\hat{\Omega }_{A,B}\), one has \(\lambda_{m}(q^{*})=0\) for some \(m\in \mathbf{N}\). As \(q^{*} \le B\), by the comparison results for eigenvalues, one has \(\lambda_{n+1}(q^{*}) \ge\lambda_{n+1}(B) >0\). Thus \(m\le n\).
Lemma 8
The number of intervals \(J_{j}\) ’s in (41), including the degenerate ones, is precisely n. Moreover, by labeling \(J_{i}\) according to the order in R, one has \(J_{i}\subset(t_{i1},t_{i})\) for \(i=1,2,\ldots, n\), where \(t_{i}\) ’s are as in Lemma 7.
Proof
Step 1. For each \(i=1,2,\ldots, n\), we assert that \((t_{i1},t_{i})\) contains at most one interval \(J_{j}\) from (41).
Otherwise, we would have two neighboring intervals \([\xi,\eta]\) and \([\hat{\xi},\hat{\eta}]\) from \(J_{j}\)’s such that \(t_{i1} < \xi\le\eta < \hat{\xi}\le\hat{\eta}<t_{i}\) and \(y(t)< c_{*}\) on \((\eta,\hat{\xi})\). Since \(y(t)\) does not change sign in \((t_{i1},t_{i})\), let us assume that \(y(\eta)=y(\hat{\xi})=c_{*}\) and \(0< y(t)< c_{*}\) on \((\eta,\hat{\xi})\). Then, on \((\eta,\hat{\xi})\), \(\ddot{y}(t) =  q^{*}(t) y(t) =  B y(t) <0\) and \(y(t)\) is strictly concave. Hence we would have \(y(t) >c_{*}\) on \((\eta,\hat{\xi})\), a contradiction.
Step 2. Let \(J_{j}=[\xi,\eta]\) be any interval from (41) such that \(J_{j}=[\xi,\eta]\subset(t_{i1},t_{i})\). For \(t\in[\eta,t_{i}]\), \(y(t)\) is given by (40). Thus \(c_{*}=y(\eta) \le1/\sqrt{B}\).
Step 3. For each \(i=1,2,\ldots, n\), we assert that \((t_{i1},t_{i})\) contains precisely one interval \(J_{j}\) from (41).
Otherwise, from Step 1, let us assume that \((t_{i1},t_{i})\) contains no \(J_{j}\) from (41). Thus \(y(t)< c_{*}\) on \((t_{i1},t_{i})\). On \([t_{i1},t_{i}]\), \(y(t)\) is given by (40). As \(y(t_{i1})=0\), we have from (40) that \(t_{i}t_{i1}=\pi/\sqrt{B}\). Thus \(c_{*}> y((t_{i1}+t_{i})/2)=1/\sqrt{B}\), a contradiction with the result on \(c_{*}\) in Step 2. □
In order to construct \(q^{*}(t)\) and \(y(t)\) on J, we need to distinguish two cases.
Next, for any i, it follows from (43) and (44) that \(J_{i}=[\xi_{i},\eta_{i}]\) has the same length with \(J_{1}\). Hence \(t_{i}t_{i1}=1/n\) for all i, i.e., \(t_{i}=i/n\) for \(i=0,1,\ldots,n\). By using the parameter τ in (44), one has then \(J_{i}=1/n2\tau \).
Lemma 9
Lemma 10
Here formulas (55) and (56) are deduced from (53) and from (42), (54), respectively.
Proof of Theorem 3
Moreover, the optimality of \(r(A,B)\) follows from Lemmas 5 and 6, by simply taking \(\hat{q}=q^{*}\).
Case 3: \(A=\infty \). In this case, one has \(\Omega _{\infty ,B} \supset \Omega _{A,B}\) for any \(A\in(\infty ,0]\). By the meaning of \(r(\infty ,B)\), one has \(r(\infty ,B) \ge r(0,B) = n\pi\sqrt{B}\).
Remark 11
4 Nondegenerate potentials for the Neumann problem
By the approach in the preceding sections, for the Neumann problem, the results are as follows.
Theorem 12
Let B and A be as in (6) and (7), where \(n\in \mathbf{N}\). By letting \(r(A,B)\) and \(\check{\Omega }_{A,B}\) be as those for the Dirichlet problem, any \(q\in\check{\Omega }_{A,B}\) is also nondegenerate with respect to problem (1)(61).
Theorem 13
Proof
As for the optimality, one needs only to notice that the zero potential \(q=0 \in \Omega _{A,B}\) is degenerate with respect to problem (1)(61). □
Declarations
Acknowledgements
The authors would like to thank Zhiyuan Wen for helpful discussions. This work is supported by the National Natural Science Foundation of China (Grant No. 11231001 and No. 11371213) and the National 111 Project of China (Station No. 111201).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Chen, H, Li, Y: Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities. Proc. Am. Math. Soc. 135, 39253932 (2007) MATHView ArticleGoogle Scholar
 Chen, H, Li, Y: Bifurcation and stability of periodic solutions of Duffing equations. Nonlinearity 21, 24852503 (2008) MATHMathSciNetView ArticleGoogle Scholar
 Li, Y, Wang, H: Neumann problems for second order ordinary differential equations across resonance. Z. Angew. Math. Phys. 46, 393406 (1995) MATHMathSciNetView ArticleGoogle Scholar
 Lin, Y, Li, Y, Zhou, Q: Second boundary value problems for nonlinear ordinary differential equations across resonance. Nonlinear Anal. 28, 9991009 (1997) MATHMathSciNetView ArticleGoogle Scholar
 Meng, G, Yan, P, Lin, X, Zhang, M: Nondegeneracy and periodic solutions of semilinear differential equations with deviation. Adv. Nonlinear Stud. 6, 563590 (2006) MATHMathSciNetGoogle Scholar
 Wang, H, Li, Y: Two point boundary value problems for second order ODEs across many resonant points. J. Math. Anal. Appl. 179, 6175 (1993) MATHMathSciNetView ArticleGoogle Scholar
 Wang, H, Li, Y: Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations. Z. Angew. Math. Phys. 47, 373384 (1996) MATHMathSciNetView ArticleGoogle Scholar
 Yang, X: SturmLiouville problems for second order ordinary differential equations across resonance. J. Optim. Theory Appl. 152, 814822 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Cabada, A, Cid, JA: On comparison principles for the periodic Hill’s equation. J. Lond. Math. Soc. (2) 86, 272290 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Cabada, A, Cid, JA, Tvrdy, M: A generalized antimaximum principle for the periodic one dimensional pLaplacian with sign changing potential. Nonlinear Anal. 72, 34363446 (2010) MATHMathSciNetView ArticleGoogle Scholar
 Torres, PJ, Zhang, M: A monotone iterative scheme for a nonlinear second order equation based on a generalized antimaximum principle. Math. Nachr. 251, 101107 (2003) MATHMathSciNetView ArticleGoogle Scholar
 Zhang, M: Optimal conditions for maximum and antimaximum principles of the periodic solution problem. Bound. Value Probl. 2010, Article ID 410986 (2010) View ArticleGoogle Scholar
 Kunze, M, Ortega, R: On the number of solutions to semilinear boundary value problems. Adv. Nonlinear Stud. 4, 237249 (2004) MATHMathSciNetGoogle Scholar
 Li, W, Zhang, M: Nondegeneracy and uniqueness of periodic solutions for some superlinear beam equations. Appl. Math. Lett. 22, 314319 (2009) MATHMathSciNetView ArticleGoogle Scholar
 Ortega, R, Zhang, M: Optimal bounds for bifurcation values of a superlinear periodic problem. Proc. R. Soc. Edinb., Sect. A, Math. 135, 119132 (2005) MATHMathSciNetView ArticleGoogle Scholar
 Torres, PJ, Cheng, Z, Ren, J: Nondegeneracy and uniqueness of periodic solutions for 2norder differential equations. Discrete Contin. Dyn. Syst., Ser. A 33, 21552168 (2013) MATHMathSciNetView ArticleGoogle Scholar
 Zhang, M: Certain classes of potentials for pLaplacian to be nondegenerate. Math. Nachr. 278, 18231836 (2005) MATHMathSciNetView ArticleGoogle Scholar
 Cañada, A, Montero, JA, Villegas, S: Lyapunovtype inequalities and Neumann boundary value problems at resonance. Math. Inequal. Appl. 8, 459475 (2005) MATHMathSciNetGoogle Scholar
 Cañada, A, Montero, JA, Villegas, S: Lyapunov inequalities for partial differential equations. J. Funct. Anal. 237, 176193 (2006) MATHMathSciNetView ArticleGoogle Scholar
 Cañada, A, Villegas, S: Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues. J. Eur. Math. Soc. 12, 163178 (2010) MATHView ArticleGoogle Scholar
 Zhang, M: Continuity in weak topology: higher order linear systems of ODE. Sci. China Ser. A 51, 10361058 (2008) MATHMathSciNetView ArticleGoogle Scholar
 Athans, M, Falb, PL: Optimal Control: An Introduction to the Theory and Its Applications. McGrawHill, New York (1966) Google Scholar