Differential equations, infinite-order system of

infinite system of differential equations

An infinite set of differential equations

$$ \tag{1 } \frac{d x _ {i} }{dt} = f _ {i} ( t , x _ {1} ,\dots ),\ \ i = 1 , 2 \dots $$

containing an infinite set of unknown functions $ x _ {k} ( t) $, $ k = 1 , 2 \dots $ and their derivatives. A solution of such a system is defined as a set of functions $ \{ x _ {k} ( t) \} $ for which all the equations of the system hold identically.

The system (1) is said to be countable, as distinct from the uncountable system

$$ \tag{2 } \frac{d x _ \alpha }{dt} = f _ \alpha ( t \dots x _ \alpha ,\dots ) $$

where $ \alpha $ runs through some uncountable set of values. Systems of the type (2) contain an uncountable set of functions $ \{ x _ \alpha ( t) \} $ which must be determined as well as their derivatives. Partial differential equations containing an uncountable set of unknown functions in two or more arguments are also studied.

A.N. Tikhonov [1] is the author of the first publication on the theory of systems of differential equations of the type (1). His main result was an existence proof of a solution of (1), under the assumption that its right-hand sides are defined for arbitrary values $ x _ {1} , x _ {2} \dots $ $ 0 \leq t - t _ {0} \leq a $, are continuous with respect to $ x _ {1} , x _ {2} \dots $ for a given value of $ t $, and are measurable with respect to $ t $ for given $ x _ {1} , x _ {2} \dots $ on an interval $ [ t _ {0} , t _ {0} + a ] $. If, in addition, the generalized Lipschitz conditions

$$ | f _ {n} ( t , x _ {1} ^ \prime , x _ {2} ^ \prime ,\dots ) - f _ {n} ( t , x _ {1} ^ {\prime \prime } , x _ {2} ^ {\prime \prime } ,\dots ) | \leq \ \sum _ {i = 1 } ^ \infty K _ {n _ {i} } | x _ {i} ^ \prime - x _ {i} ^ {\prime \prime } | $$

are met, and the series

$$ \sum _ {n = 1 } ^ \infty K _ {n _ {i} } = A _ {i} < A $$

converge and are uniformly bounded, then the solution $ x _ {i} ( t) $, $ i = 1 , 2 \dots $ of (1) is unique if the given initial conditions are such that the series

$$ \sum _ {n = 1 } ^ \infty | x _ {n} ( t) | $$

converges.

The subsequent development of the theory of countable systems concerned the conditions of boundedness of the solutions [2], analytic dependence on the parameter, Lyapunov stability and other properties of the solutions [2]. Linear and quasi-linear countable systems of differential equations have been particularly thoroughly studied.

Operator methods are especially fruitful in the study of systems of infinite order. For instance, rather than the system (1) one considers the operator equation

$$ \tag{3 } \frac{dx}{dt} = f ( t , x ) , $$

where $ x ( t) $ is an infinite-dimensional vector of some Banach space $ B $, $ f ( t , x ) $ is an infinite-dimensional vector-function with values in this space, and the derivative is considered in the sense of Fréchet. In particular, the following results were obtained [3] for equation (3).

If $ f ( t , x ) $ is a bounded operator, it follows from the validity of the local existence theorem that if the Bohl exponent at zero is negative [3], then the solutions with initial values close to zero can be defined on arbitrary large intervals.

If

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

where $ A $ is a bounded operator given by an infinite-dimensional matrix, then all solutions are bounded for $ - \infty < t < \infty $ in a Hilbert space if and only if $ A $ is similar to a skew-Hermitian matrix. In this case, the explicit form of the solution of the Cauchy problem with initial conditions $ x ( t _ {0} ) = x _ {0} $ was found in the form

$$ x ( t) = e ^ {A ( t - t _ {0} ) } x _ {0} , $$

where $ e ^ {At} $ is the operator exponent.

If

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

where $ A $ has its previous meaning while $ f ( t) $ is a continuous $ T $- periodic vector-function, there exists a unique periodic solution when the spectrum $ \sigma ( A) $ of $ A $ does not contain the points $ 2 k \pi i / T $, $ k = 0, \pm 1 \dots $ of the imaginary axis.

Conditions for Lyapunov stability at zero were found for the case

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

in the form of the requirement

$$ \| F ( t , x ) \| \langle q \| x \| ,\ t \rangle 0 ,\ \ \| x \| < \rho , $$

for a sufficiently small $ q > 0 $, if the spectrum of $ A $ lies in the open left half-space.

References

[1]	A.N. Tikhonov, Mat. Sb. , 41 : 4 (1934) pp. 551–560
[2]	K.G. Valeev, O.A. Zhautykov, "Infinite systems of differential equations" , Alma-Ata (1974) (In Russian)
[3]	Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)

Comments

Consider a homogeneous differential equation

$$ \tag{a1 } \frac{dx}{dt} = A ( t) x ,\ \ 0 \leq t < \infty , $$

where $ A ( t) $ is an operator-valued function, $ A ( t) $ an operator on the Banach space $ B $, $ x \in B $. Let

$$ x ( t) = u ( t) x _ {0} $$

be a solution, $ x ( 0) = x _ {0} $. The (upper) Bohl exponent $ K _ {B} ( x _ {0} ) $ of this solution is the greatest lower bound of all real numbers $ \rho $ such that there exists an $ N _ \rho $ for which

$$ \| x ( t) \| \leq N _ \rho \mathop{\rm exp} ( \rho ( t - \tau ) ) \| x ( t) \| $$

for all $ 0 \leq \tau \leq t < \infty $. The lower Bohl exponent $ K _ {B} ^ { \prime } ( x _ {0} ) $ is the least upper bound of the numbers $ \lambda $ for which there exists an $ M _ \lambda > 0 $ such that

$$ \| x ( t) \| \geq M _ \lambda \mathop{\rm exp} ( x ( t - \tau ) ) \| x ( t) \| . $$

If $ \lambda ( x _ {0} ) $ is a Lyapunov exponent of (a1), cf. Lyapunov characteristic exponent, then

$$ - \infty \leq K _ {B} ^ { \prime } ( x _ {0} ) \leq \ \lambda ( x _ {0} ) \leq K _ {B} ( x _ {0} ) \leq \infty . $$

The interval $ [ K _ {B} ^ { \prime } ( x _ {0} ) , K _ {B} ( x _ {0} ) ] $ is called the Bohl interval of the solution in question.

Now consider again the equation (3) and let $ f ( t , 0 ) = 0 $. This equation is said to satisfy property $ {\mathcal B} ( \nu , N , \rho ) $, $ - \infty < \nu < \infty $, $ N > 0 $, $ \rho > 0 $, if every solution of it with $ \| x ( t _ {0} ) \| \leq \rho $ at some time $ t _ {0} $, satisfies the estimate

$$ \| x ( t) \| \leq N \mathop{\rm exp} ( - \nu ( t - \tau ) ) \| x ( \tau ) \| $$

for all $ t > \tau \geq t _ {0} $ for which the solution is defined. Generalizing the definitions above, the (upper) Bohl exponent at zero is the greatest lower bound of the $ \lambda = - \nu $ for which there exist $ N _ \nu $, $ \rho _ \nu $ such that the equation has property $ {\mathcal B} ( \nu , N _ \nu , \rho _ \nu ) $.

References

[a1]	A. Pazy, "Differential equations in Banach spaces" , Springer (To appear)