Boundary value problem, ordinary differential equations

The problem of finding a solution to an equation

$$ \tag{1 } \frac{dx }{dt } = \ f (t, x),\ \ t \in J,\ \ x \in \mathbf R ^ {n} , $$

lying in a given subset $ D $ of the space $ D (J, \mathbf R ^ {n} ) $ of functions depending on $ t $ that are absolutely continuous on $ J $ and that assume values in $ \mathbf R ^ {n} $:

$$ \tag{2 } x ( \cdot ) \in D. $$

It is assumed that $ f (t, x) $ is a function defined on $ J \times \mathbf R ^ {n} $ with values in $ \mathbf R ^ {n} $ and satisfying the Carathéodory conditions; $ J $ is an interval on the real line $ \mathbf R $.

1) The boundary value problem (1), (2) is said to be linear if

$$ f (t, x) \equiv \ A (t) x + b (t), $$

where the functions $ A (t) $ and $ b (t) $ are summable on every compact interval in $ J $ and the set $ D $ is a linear manifold in $ D (J, \mathbf R ^ {n} ) $. In particular, one might have

$$ J = \ [t _ {0} , t _ {1} ], $$

$$ D = \left \{ x ( \cdot ) \in D (J, \mathbf R ^ {n} ): \int\limits _ { t _ {0} } ^ { {t _ 1 } } [d \Phi (t)] x (t) = 0 \right \} , $$

where $ \Phi (t) $ is a function of bounded variation. A linear boundary value problem gives rise to a linear operator

$$ Lx (t) \equiv \ x ^ \prime - A (t) x,\ \ x ( \cdot ) \in D, $$

the eigen values of which are precisely those values of the parameter $ \lambda $ for which the homogeneous boundary value problem

$$ x ^ \prime - A (t) x = \lambda x,\ \ x ( \cdot ) \in D, $$

has non-trivial solutions. These non-trivial solutions are the eigen functions of the operator $ L $. If the inverse operator $ L ^ {-1} $ exists and has an integral representation

$$ x (t) = \ L ^ {-1} b (t) \equiv \ \int\limits _ { J } G (t, s) b (s) ds,\ \ t \in J, $$

then $ G (t, s) $ is called a Green function.

2) Let $ J = (- \infty , \infty ) $, let $ f (t, x) $ be almost-periodic in $ t $ uniformly in $ x $ on every compact subset of $ \mathbf R ^ {n} $ and let $ D $ be the set of almost-periodic functions in $ t $ that are absolutely continuous on $ J $. Then problem (1), (2) is known as the problem of almost-periodic solutions.

3) In control theory one considers boundary value problems with a functional parameter: a control. For example, consider the equation

$$ \tag{3 } \frac{dx }{dt } = \ f (t, x, u),\ \ t \in J = [t _ {0} , t _ {1} ],\ \ x \in \mathbf R ^ {n} , $$

with set of admissible controls $ U $ and two sets $ M _ {0} , M _ {1} \subset \mathbf R ^ {n} $. Let $ D $ be the set of absolutely continuous functions in $ t $ such that $ x (t _ {0} ) \in M _ {0} $, $ x (t _ {1} ) \in M _ {1} $. The boundary value problem is to find a pair $ (x _ {0} ( \cdot ), u _ {0} ( \cdot )) $ such that $ u _ {0} ( \cdot ) \in U $ and the solution $ x _ {0} (t) $ of equation (3) at $ u = u _ {0} (t) $ satisfies the condition $ x _ {0} ( \cdot ) \in D $.

4) There is a wide range of diverse necessary and sufficient conditions for the existence and uniqueness of solutions to various boundary value problems, and of methods for constructing an approximate solution (see [4]–[7]). For example, consider the problem

$$ \tag{4 } \left . \begin{array}{c} x ^ \prime = A (t) x + f (t, x), \\ \int\limits _ { t _ {0} } ^ { {t _ 1 } } [d \Phi (t)] x (t) = 0, \end{array} \ \right \} $$

in which

$$ \| f (t, x) \| \leq a + b \ \| x \| ^ \alpha $$

for certain constants $ a > 0, b > 0, \alpha \geq 0 $. Suppose that the homogeneous problem

$$ \tag{5 } x ^ \prime = A (t) x,\ \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } [d \Phi (t)] x (t) = 0 $$

is regular, i.e. its only solution is the trivial one. Then problem (4) has at least one solution, provided either $ \alpha < 1 $, or $ \alpha \geq 1 $ and $ b $ is sufficiently small. It is fairly complicated to determine whether problem (5) is regular. However, the linear (scalar) boundary value problem

$$ x ^ {\prime\prime} + q (t) x ^ \prime + p (t) x = 0,\ \ x (t _ {0} ) = 0,\ \ x (t _ {1} ) = 0, $$

for example, is regular if whenever $ | q (t) | \leq 2m $ there exists a $ k \in \mathbf R $ such that

$$ \int\limits _ { t _ {0} } ^ { {t _ 1 } } [p (t) - k] _ {+} dt < \ 2 [F (k, m) - m], $$

where

$$ F (k, m) = \ \left \{ \begin{array}{l} \sqrt {k - m ^ {2} } \mathop{\rm cotg} \ { \frac{(t _ {1} - t _ {0} ) \sqrt {k - m ^ {2} } }{2} } , \\ \ \ m ^ {2} < k \leq m ^ {2} + \frac{\pi ^ {2} }{(t _ {1} - t _ {0} ) ^ {2} } , \\ \frac{2}{t _ {1} - t _ {0} } ,\ \ k = m ^ {2} , \\ \sqrt {m ^ {2} - k } \ \mathop{\rm cotg} \ { \frac{(t _ {1} - t _ {0} ) \sqrt {m ^ {2} - k } }{2} } ,\ \ k < m ^ {2} . \\ \end{array} \right .$$

References

[1]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[2]	M.A. Krasnosel'skii, V.Sh. Burd, Yu.S. Kolesov, "Nonlinear almost-periodic oscillations" , Wiley (1973) (Translated from Russian)
[3]	L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Interscience (1962) (Translated from Russian)
[4]	N.N. Krasovskii, "Theory of control of motion. Linear systems" , Moscow (1968) (In Russian)
[5]	V.I. Zubov, "Lectures in control theory" , Moscow (1975) (In Russian)
[6]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[7]	I.T. Kiguradze, "Some singular boundary value problems for ordinary differential equations" , Tbilisi (1975) (In Russian)

References

[a1]	M. Braun, "Differential equations and their applications" , Springer (1975)
[a2]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) pp. §§3.6, 3.51, 4.7, A.5
[a3]	L.K. Jackson, "Boundary value problems for ordinary differential equations" J.K. Hale (ed.) , Studies in ordinary differential equations , Math. Assoc. Amer. (1977) pp. 93–127