Delay and Functional Differential Equations

Introduction

The equations of the preceding article determine the derivative of the unknown at a time $t$ from its value at the same time $t$. Many of the equations of analysis and of the sciences are not of this kind: the rate of change at a time depends on the state at an earlier time, or on a whole segment of earlier states. The simplest instance is the delay differential equation $y'(t) = f(t, y(t), y(t-r))$ with a fixed lag $r > 0$, and the general instance is the functional differential equation $y'(t) = f(t, y_t)$, where $y_t$ is the whole history of $y$ over the interval $[t-r,t]$ and the equation is a relation between a finite-dimensional vector and an infinite-dimensional datum. This article develops the theory of that class.

The two features that distinguish it from the ordinary theory are structural and govern everything below. First, an initial condition is no longer a point but a function on an interval — a history — so the solution space is infinite-dimensional and the natural setting is a space of continuous (or integrable) functions on a compact interval. Second, the Picard iteration of the ordinary theory still works, but only after the equation has been integrated step by step, because the equation on an interval of length $r$ refers to data already known; this is the method of steps, and it is what makes a delay equation an initial-value problem in a space of histories rather than a boundary-value problem.

The setting is a Banach space $X$ over $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ as in Banach and Hilbert Spaces, a fixed lag $r > 0$, and the history space $C = C([-r,0],X)$ with the supremum norm. The ordinary theory of the preceding article is assumed throughout, and the notation for the Picard operator, the maximal interval and the flow is that article's. The delay equations are the first and most important case of the functional equations, and the article follows that order: the method of steps, existence and uniqueness for a history, the linear theory with its transcendental characteristic equation, the solution operator and the variation-of-constants formula, the stability theory by Lyapunov functionals and the Razumikhin method, and the general functional equation with its own existence theorem. The qualitative theory of the solution operator as a one-parameter semigroup, and its spectral decomposition, belongs with the evolution equations of this Part and is stated here only in its elementary form.

Delay Equations and the History

Definitions

Definition. Let $r > 0$ and let $f : \mathbb{R} \times X \times X \to X$ be a function. The delay differential equation with lag $r$ determined by $f$ is

$$ y'(t) = f\bigl(t, y(t), y(t-r)\bigr) . $$

It is retarded if the derivative depends on the past and not on a future value, neutral if it depends on $y'(t-r)$ as well, and advanced if it depends on a future value $y(t+r)$. The equation has constant lag $r$ when $r$ is a constant, proportional or variable lag when the lag is a function $r(t) \ge 0$, and state-dependent when the lag depends on $y$.

Definition. A history for the equation on an interval $[\sigma, \sigma + \cdot\,]$ is a continuous function $\phi : [-r,0] \to X$, and the initial-value problem with initial time $\sigma$ and history $\phi$ is the equation together with

$$ y(\sigma + \theta) = \phi(\theta) \qquad (\theta \in [-r,0]). $$

A solution on $[\sigma-r, \sigma+T)$ is a continuous function $y$ that is differentiable on $(\sigma, \sigma+T)$, agrees with $\phi$ on $[\sigma-r,\sigma]$, and satisfies the equation for $t > \sigma$. The history space is $C = C([-r,0],X)$, a Banach space under $\|\phi\|_\infty = \sup_{\theta}|\phi(\theta)|$.

The distinction between an initial point and an initial history is the whole difference between the ordinary and the functional theory. A delay equation is not a well-posed initial-value problem at a point: two solutions through the same point but with different histories generally differ for all later times. In compensation, the history determines the solution uniquely, exactly as the point does for an ordinary equation.

Proposition (smoothness propagation). If $f$ is of class $C^k$ and the history $\phi$ is of class $C^k$ on $[-r,0]$ satisfying the compatibility conditions at the initial time $\sigma$,

$$ \phi^{(j)}(0^-) = \frac{d^{j}}{dt^{j}} f\bigl(t, y(t), y(t-r)\bigr)\Bigr|_{t=\sigma^-} \qquad (j < k), $$

then the solution is of class $C^k$ on $[\sigma-r,\sigma+T)$; without compatibility the solution is only continuous at $\sigma$, with a jump in the derivative at $\sigma$.

Proof. The solution is differentiable for $t>\sigma$ by the equation, and its derivative at the initial time is the prescribed limit only when $\phi$ is compatible with the right-hand side, since the right-hand side at $t = \sigma$ is computed from $\phi$; each further differentiation of the equation raises the order of the condition by one and requires the corresponding derivative of $\phi$ to exist and match.

Example (discontinuous derivative). For $y'(t) = -y(t-1)$ with history $\phi(\theta) = 1$ for $\theta \in [-1,0]$, the solution is $\phi$ continued by $y(t) = 1 - t$ on $[0,1]$, then $y(t) = 1 - t + \tfrac12(t-1)^2$ on $[1,2]$, and so on. The derivative jumps at $t = 0$ from $0$ to $-1$, because $\phi'(0^-) = 0$ does not match the equation. The solution becomes smoother with each step, a smoothing that is invisible to a point initial condition.

The Method of Steps

Theorem (method of steps). Let $\sigma \in \mathbb{R}$, let $\phi \in C([-r,0],X)$, and let $f : \mathbb{R}\times X\times X \to X$ be continuous and locally Lipschitz in its second and third variables. Then there is $T > 0$ and a unique solution on $[\sigma-r,\sigma+T]$.

Proof. On the interval $[\sigma,\sigma+r]$ the values $y(t-r)$ for $t \in [\sigma,\sigma+r]$ lie in $[\sigma-r,\sigma]$ and are given by the history, so the equation is the ordinary initial-value problem

$$ y'(t) = f\bigl(t, y(t), \phi(t-\sigma-r)\bigr), \qquad y(\sigma) = \phi(0), $$

whose right-hand side is continuous and locally Lipschitz in $y$; the Picard–Lindelöf theorem of the preceding article gives a unique solution on $[\sigma, \sigma+\delta_1]$ where $\delta_1 > 0$ depends only on the bounds of $f$ and $\phi$. If $\delta_1 \ge r$ the first step is complete; otherwise the argument restarts at $\sigma+\delta_1$. In either case one obtains a solution on $[\sigma,\sigma+r]$. On the next interval $[\sigma+r, \sigma+2r]$ the values $y(t-r)$ lie in $[\sigma,\sigma+r]$ and are now the known solution of the first step, so the same argument applies, and iteration over successive intervals of length $r$ gives the solution. The solution is unique on each step because the ordinary problem is, and the steps cover all of $[\sigma, \sigma+T]$ for every $T$ for which the bounds allow.

The method of steps is not merely a proof device: it is the algorithm by which the solution is computed, and it shows that a delay equation is an ordinary equation repeating with the data of the preceding step.

Corollary (maximal interval). The solution extends to a maximal interval $[\sigma-r,\beta)$ with $\beta > \sigma$; if $\beta < \infty$ then the solution leaves every compact subset of $X$ as $t \to \beta^-$.

Proof. Each step is an ordinary initial-value problem, to which the maximal-interval theorem for the ordinary theory applies; the maximal interval of the delay equation is the union of the maximal intervals of the steps.

Example (logistic delay). The equation $y'(t) = y(t)\bigl(1 - y(t-1)\bigr)$ with history $\phi \equiv \tfrac12$ is solved on $[0,1]$ by $y' = y(1-\tfrac12)$, that is $y(t) = \tfrac12 e^{t/2}$; the second step solves a logistic equation with a known forcing term, and the iteration continues. The solution remains defined for all time but its large-time behaviour depends on the lag, as the linear analysis below explains.

Linear Delay Equations

The Characteristic Equation

Definition. A linear delay equation with constant coefficients is

$$ y'(t) = A\,y(t) + B\,y(t-r), $$

with $A, B \in B(X)$ bounded operators, and it is homogeneous; the general linear equation has an additional term $b(t)$.

Theorem (exponential solutions). The scalar equation $y'(t) = ay(t) + by(t-r)$ has a solution $y(t) = e^{\lambda t}$ if and only if

$$ \lambda = a + b\,e^{-\lambda r} , $$

the characteristic equation. This transcendental equation has countably many roots $\lambda_n$, none of them with a finite accumulation point in $\mathbb{C}$, and for each root $e^{\lambda t}$ is a solution; when a root has multiplicity $m$, the functions $t^{k}e^{\lambda t}$, $0 \le k < m$, are also solutions.

Proof. Substitution gives $\lambda e^{\lambda t} = (a + be^{-\lambda r})e^{\lambda t}$; the equation in $\lambda$ is transcendental and therefore has countably many roots. If $\lambda$ has multiplicity $m$ as a root of $\lambda - a - be^{-\lambda r}$, differentiation of the identity $m-1$ times with respect to $\lambda$ shows that $t^k e^{\lambda t}$ solves the equation for $k < m$.

The contrast with the ordinary theory is exact: a scalar linear ordinary equation has $n$ roots and an $n$-dimensional solution space, whereas a scalar linear delay equation has infinitely many roots and an infinite-dimensional solution space, because its initial datum is a function. The spectrum of the solution operator is the set of characteristic roots, and it is discrete but infinite.

Example (the pure delay). For $y'(t) = -a\,y(t-r)$ with $a > 0$ the characteristic equation is $\lambda = -ae^{-\lambda r}$. Because $|\lambda| = a e^{-r\operatorname{Re}\lambda}$, every root satisfies $\operatorname{Re}\lambda \le a$, and the roots with $\operatorname{Re}\lambda\ge0$ satisfy $|\lambda|\le a$; throughout the spectrum one has $|\lambda| = ae^{-r\operatorname{Re}\lambda}\le ae^{ar}$, so the roots lie in the disc $\{|\lambda|\le ae^{ar}\}$, and the number of roots in any half-plane $\{\operatorname{Re}\lambda > \gamma\}$ is finite. The roots depend continuously on $a$ and $r$, and a root crosses the imaginary axis precisely when

$$ i\omega = -a e^{-i\omega r}, $$

which forces $a = \omega$ by taking moduli and $\omega r = \pi/2 + 2k\pi$ by taking arguments. Hence the stability condition is $ar < \pi/2$: all characteristic roots have strictly negative real part, and every solution decays, exactly when $ar < \pi/2$. At $ar = \pi/2$ there is a pair of purely imaginary roots $\pm i\omega$ with $\omega = a$, and for $ar > \pi/2$ a pair has crossed to the right half-plane and the zero solution is unstable.

The example shows that a delay can destabilise a system that is stable without it: $-ay(t)$ is stable for every $a>0$, but $-ay(t-r)$ is stable only when the product of gain and lag is below $\pi/2$.

The Solution Operator and Variation of Constants

Definition. The solution operator of the homogeneous linear delay equation at time $t \ge 0$ is the bounded operator $T(t) : C \to C$ sending a history $\phi \in C$ to the history of the solution at time $t$, that is $T(t)\phi = y_t$ where $y_t(\theta) = y(t+\theta)$ for $\theta \in [-r,0]$.

Theorem (semigroup property). The solution operators satisfy $T(0) = I$, $T(t+s) = T(t)\,T(s)$ for $s, t \ge 0$, and $t \mapsto T(t)\phi$ is continuous from $[0,\infty)$ to $C$ for every $\phi \in C$; moreover $\|T(t)\|$ is bounded on each compact interval, and $T(t)$ is compact for $t \ge r$.

Proof. The semigroup property is the uniqueness of the solution: the history of the solution at $t+s$ is the history of the solution starting from the history at $t$, and the initial-value problem with a history in $C$ has a unique solution. Strong continuity follows from the continuity of solutions in their initial history, which is the estimate of the next theorem. The compactness for $t \ge r$ is the Arzelà–Ascoli theorem: the operator $T(t)$ maps bounded subsets of $C$ to subsets that are uniformly bounded and equicontinuous, since the solution is Lipschitz on $[t-r,t]$ with a bound depending only on the norm of the history.

The semigroup property places linear delay equations inside the theory of strongly continuous semigroups of bounded operators, which is developed for general evolution equations; here it is used only to state the variation-of-constants formula, and the spectral theory of $T(t)$ is the spectral theory of the characteristic equation.

Theorem (variation of constants for a delay equation). Let $y$ solve the inhomogeneous equation $y'(t) = Ay(t) + By(t-r) + b(t)$ on $[\sigma,\infty)$ with history $\phi$ and with $b$ continuous. Then

$$ y(t) = \bigl(T(t-\sigma)\phi\bigr)(0) + \int_{\sigma}^{t} K(t-s)\,b(s)\,ds , $$

where $K$ is the fundamental solution: the solution of the homogeneous equation with the history $K(\theta) = 0$ for $\theta<0$ and the jump $K(0)=I$, continued by the equation. Equivalently, in the scalar case,

$$ K(t) = \frac{1}{2\pi i}\int_{\Gamma} \frac{e^{zt}}{z - a - be^{-zr}}\,dz , $$

the contour $\Gamma$ enclosing all characteristic roots to the right of a vertical line, the integral being an inverse Laplace transform.

Proof. By linearity it suffices to treat the two terms separately. The first is the homogeneous solution. For the second, both sides satisfy the inhomogeneous equation and vanish for $t \le \sigma$, by the causality of the integral and the vanishing of the history of $K$; uniqueness of the solution of the initial-value problem completes the argument. The contour formula is the inversion of the Laplace transform, valid to the right of the rightmost characteristic root.

Theorem (continuous dependence). With $A, B$ bounded and $\|A\| + \|B\| \le M$, the solution of the homogeneous equation satisfies

$$ \|y_t\|_\infty \le \|\phi\|_\infty\, e^{Mt} \qquad (t \ge 0). $$

Proof. The integral equation $y(t) = \phi(0) + \int_\sigma^t(Ay(s)+By(s-r))\,ds$ gives, for the supremum $u(t) = \sup_{\sigma-r\le s\le t}\|y(s)\|$, the inequality $u(t) \le \|\phi\|_\infty + M\int_\sigma^t u(s)\,ds$; Gronwall's inequality of the preceding article gives the bound.

Stability and Lyapunov Functionals

Asymptotic Stability

Definition. The zero solution of a linear delay equation is exponentially stable if there are constants $C \ge 1$ and $\gamma > 0$ with $\|y_t\|_\infty \le C e^{-\gamma t}\|\phi\|_\infty$ for every history $\phi$, and asymptotically stable if $y_t \to 0$ for every $\phi$. For the nonlinear equation $y'(t) = f(t,y(t),y(t-r))$ with $f(t,0,0) = 0$, the zero solution is stable if for every $\varepsilon > 0$ there is $\delta > 0$ with $\|\phi\|_\infty < \delta$ implying $\|y_t\|_\infty < \varepsilon$ for all $t \ge 0$, and asymptotically stable if in addition $y_t \to 0$.

Theorem (spectral criterion). The linear equation $y'(t) = Ay(t) + By(t-r)$ is exponentially stable if and only if every characteristic root $\lambda$ of

$$ \det\bigl(\lambda I - A - Be^{-\lambda r}\bigr) = 0 $$

satisfies $\operatorname{Re}\lambda \le -\gamma$ for some $\gamma > 0$; equivalently, if and only if all roots lie in a fixed left half-plane.

Proof. From the contour representation of the fundamental solution, the growth of $K(t)$ as $t\to\infty$ is governed by the rightmost characteristic root, and the estimate $|e^{\lambda t}| = e^{(\operatorname{Re}\lambda)t}$ gives the equivalence; the passage from the scalar contour formula to the matrix determinant is the standard Laplace-transform solution of the linear system.

Example. For $y'(t) = -ay(t-r)$ the criterion reproduces the condition $ar < \pi/2$ found above. For $y'(t) = -ay(t) - by(t-r)$ the boundary of stability in the $(a,b)$-plane is traced by the purely imaginary roots $\lambda = i\omega$. Separating the real and imaginary parts of $i\omega = -a - b(\cos\omega r - i\sin\omega r)$ gives

$$ a = -\omega\cot(\omega r), \qquad b = \frac{\omega}{\sin(\omega r)} , $$

so each pair of roots crossing the imaginary axis contributes one branch, parametrised by $\omega > 0$ with $\sin(\omega r) \neq 0$. As $\omega \to 0^+$ the branch approaches $(a,b) = (-1/r, 1/r)$, and the branches accumulate at $\omega r = \pi, 2\pi, \dots$, cutting the plane into regions. Stability can change only when a root crosses the imaginary axis, so the stable region is the connected component of the complement of these branches containing the point $(a,b) = (1,0)$, which is the stable case of the ordinary equation $y' = -ay$. When $a > |b|$ the quadratic Lyapunov functional above already gives stability, so the stable component contains the half-plane $a > |b|$ and is not the left half-plane of the ordinary theory: the branches carve it into a more complicated region, and the computation of the branches is exact.

Lyapunov Functionals and Razumikhin

The spectral criterion is available only for linear equations with constant coefficients. For a nonlinear equation one uses a functional that decreases along solutions, and the two standard tools are the Lyapunov functional and the Razumikhin condition.

Definition. A Lyapunov functional for the equation $y'(t) = f(t,y_t)$ is a continuous functional $V : \mathbb{R}\times C \to \mathbb{R}$ that is positive definite, $V(t,0)=0$ and $V(t,\phi) \ge w(\|\phi(0)\|)$ for a continuous increasing $w$ with $w(0)=0$, and whose right derivative along solutions,

$$ \dot V(t,\phi) = \limsup_{h\to0^+}\frac{V\bigl(t+h, y_{t+h}(t,\phi)\bigr) - V(t,\phi)}{h}, $$

is nonpositive, where $y_{t+h}(t,\phi)$ is the history of the solution starting from $\phi$ at time $t$.

Theorem (Lyapunov stability). If a Lyapunov functional exists with $\dot V \le 0$ for all $(t,\phi)$, the zero solution is stable; if in addition $\dot V \le -w_1(\|\phi(0)\|)$ for a positive definite $w_1$ and $V$ is bounded above on bounded sets by $w_2(\|\phi\|_\infty)$, the zero solution is asymptotically stable.

Proof. This is the direct method of Lyapunov, transplanted from finite dimensions: the monotonicity of $t \mapsto V(t,y_t)$ confines the solution to sublevel sets of $V$, which are neighbourhoods of the origin by positive definiteness, and the stronger decay condition forces the solution to the origin, the boundedness of $V$ on bounded sets providing the uniformity in the initial history.

Example (a quadratic functional). For $y'(t) = -ay(t) + by(t-r)$ with $a > 0$ and $|b| < a$, the functional

$$ V(\phi) = \phi(0)^2 + \mu\int_{-r}^{0}\phi(\theta)^2\,d\theta $$

has derivative along a solution equal to $-2a\phi(0)^2 + 2b\phi(0)\phi(-r) + \mu\bigl(\phi(0)^2 - \phi(-r)^2\bigr)$ up to the boundary term of the integral, and choosing $\mu$ large enough makes the expression negative definite; hence the zero solution is asymptotically stable. The condition $|b|

Theorem (Razumikhin). Suppose there is a continuous $V : \mathbb{R}\times X \to \mathbb{R}$ with $V(t,0)=0$, $V$ positive definite, and $\dot V(t,y(t)) \le -w(\|y(t)\|)$ whenever $V(t+\theta, y(t+\theta)) \le V(t,y(t))$ for all $\theta \in [-r,0]$, where $\dot V$ is the derivative along the solution of the associated ordinary equation. Then the zero solution is asymptotically stable.

Proof. If $V(t,y(t))$ were to increase at some time, there would be, by continuity and by the condition of the history at that time, a first time at which $V(t,y(t))$ equals its previous maximum over $[t-r,t]$; at that time the hypothesis applies and gives $\dot V < 0$, a contradiction. Hence $V(t,y(t))$ is nonincreasing, and the argument proceeds as in the direct method.

Razumikhin's method avoids the integral over the history, at the cost of a hypothesis on the whole history only at the instants at which the maximum is attained; it is the more flexible tool for a nonlinearity that depends on the past through a bounded function.

General Functional Differential Equations

The General Form

Definition. Let $f : \mathbb{R} \times C \to X$ be a function. The functional differential equation determined by $f$ is

$$ y'(t) = f(t, y_t), \qquad y_t(\theta) = y(t+\theta),\quad \theta \in [-r,0]. $$

The delay equation $y'(t) = g(t,y(t),y(t-r))$ is the case $f(t,\phi) = g(t,\phi(0),\phi(-r))$, and a Volterra functional equation has $f$ depending only on $\phi$ restricted to a subinterval $[-r_0,0]$ with $r_0 \le r$. The equation is autonomous if $f$ does not depend on $t$.

Definition. $f$ is Lipschitz in the second variable with constant $L$ if $\|f(t,\phi) - f(t,\psi)\| \le L\|\phi-\psi\|_\infty$ for all $t,\phi,\psi$, and locally Lipschitz near a point if the estimate holds in a neighbourhood of it. $f$ is completely continuous if it maps bounded sets to relatively compact sets.

Theorem (existence and uniqueness). Let $f : \mathbb{R}\times C \to X$ be continuous and locally Lipschitz in its second variable. Then for every $(\sigma,\phi) \in \mathbb{R}\times C$ there is $T>0$ and a unique solution of $y'(t) = f(t,y_t)$ on $[\sigma-r,\sigma+T]$ with history $\phi$.

Proof. Write the equation as the integral equation

$$ y(t) = \phi(0) + \int_\sigma^t f(s,y_s)\,ds \qquad (t \ge \sigma), $$

on the space of continuous $y : [\sigma-r,\sigma+T] \to X$ extending $\phi$, with the supremum norm. The map $T(y)(t) = \phi(0) + \int_\sigma^t f(s,y_s)\,ds$ is a contraction on the closed ball of radius $b$ for $T$ small enough, exactly as in the proof of Picard–Lindelöf, provided the evaluation $y \mapsto y_s$ is continuous, which it is with $\|y_s - z_s\|_\infty \le \|y-z\|_\infty$; the contraction mapping principle gives a fixed point, and the fixed point is a solution by the fundamental theorem of calculus.

The proof is the Picard argument in the Banach space of histories; the only new ingredient is that the right-hand side is a function of an element of $C$ rather than of a point of $X$, and that the Lipschitz hypothesis is taken in the norm of $C$.

Theorem (existence without Lipschitz). If $f$ is continuous and completely continuous and maps bounded sets to bounded sets, then for every $(\sigma,\phi)$ there is at least one solution. Uniqueness may fail.

Proof. Quoted as standard. The argument uses the theory of condensing maps and the Schauder fixed point theorem: the integral operator is compact on bounded sets of the space of continuous extensions, so approximate solutions for the Lipschitz problems with $f_n \to f$ have a convergent subsequence whose limit solves the equation.

Linear Functional Equations and Decomposition of the Spectrum

Definition. A linear functional differential equation is

$$ y'(t) = L(t)\,y_t + b(t), $$

where $L(t) \in B(C,X)$ is a bounded linear map for each $t$ and $b$ is continuous. With $L$ constant and represented by a measure, $L\phi = \int_{-r}^{0}d\eta(\theta)\,\phi(\theta)$ for a finite matrix-valued measure $\eta$ of bounded variation, the equation is the general linear retarded equation with constant coefficients.

Theorem (characteristic equation). For $L\phi = \int_{-r}^0 d\eta(\theta)\phi(\theta)$ the exponential $e^{\lambda t}v$, $v \in \mathbb{C}^n$, solves the homogeneous equation if and only if

$$ \Delta(\lambda)v = 0, \qquad \Delta(\lambda) = \lambda I - \int_{-r}^{0}e^{\lambda\theta}\,d\eta(\theta) , $$

and the characteristic matrix $\Delta$ has a determinant that is an entire function of exponential type; its zeros are the characteristic roots, they are countable, and for every $\gamma \in \mathbb{R}$ there are only finitely many with $\operatorname{Re}\lambda > \gamma$. The spectral projection onto the root subspace of a finite set of roots can be written as a contour integral of the resolvent $(\lambda I - L)^{-1}$ around the roots.

Proof. Substitution gives the eigenvalue equation. The determinant of $\Delta$ is $\lambda^n$ plus terms involving $\int e^{\lambda\theta}\,d\eta$, each of which is entire of exponential type bounded by $e^{r|\lambda|}\|\eta\|$; Hadamard's theory of entire functions of exponential type gives the countability and the finiteness on each right half-plane, since the exponential growth of the determinant forces its zeros to be asymptotically contained in a union of vertical strips. The contour representation of the spectral projection is the standard Riesz projection of the resolvent.

Remark (the decomposition theorem). For a linear autonomous equation the space of solutions splits into a finite-dimensional part spanned by the exponential solutions corresponding to the finitely many roots in a right half-plane and a remainder on which the solution operator decays; this is the decomposition theorem of the theory, and it is the form in which the theory is used to reduce a delay equation to a finite-dimensional one plus a stable error. It is the seed of the semigroup theory of the evolution equations of this Part, where the same decomposition appears as the spectral decomposition of a semigroup with eventually compact orbits.

Examples and Applications

Example (a delay in a control loop). The scalar equation $y'(t) = -ay(t-r)$ has the characteristic equation $\lambda = -ae^{-\lambda r}$ and is stable exactly for $ar < \pi/2$. As the gain $a$ is increased with the lag fixed, the rightmost pair of roots crosses the imaginary axis at $a = \pi/(2r)$ with roots $\pm i\pi/(2r)$, and a periodic solution of period $4r$ is born. The computation is exact and shows that the delay converts a stable first-order system into an oscillator.

Example (the neutral equation). The equation $y'(t) = -y'(t-1)$ has solutions of the form $y(t) = e^{\lambda t}$ with $\lambda = -\lambda e^{-\lambda}$, so that $e^{-\lambda} = -1$ and $\lambda = i(2k+1)\pi$; all solutions are oscillatory and none decays faster than polynomially, because the roots are purely imaginary and simple. A neutral equation may fail to smooth the solution, in contrast with a retarded one, and its characteristic function may have roots with $\operatorname{Re}\lambda \to \infty$.

Example (a state-dependent delay). The equation $y'(t) = -y\bigl(t - r(y(t))\bigr)$ with $r \ge 0$ bounded and Lipschitz has a right-hand side $f(t,\phi) = -\phi(-r(\phi(0)))$ that is Lipschitz in $\phi$ when $r$ is, so the existence theorem applies; the history space must be enlarged to the Lipschitz functions when $r$ is itself state-dependent, because the evaluation at $-r(\phi(0))$ is not continuous in the supremum norm alone. This is the technical price of a state-dependent lag and it has no analogue in the constant-lag theory.

Summary

A delay differential equation $y'(t) = f(t,y(t),y(t-r))$ is posed with a history $\phi \in C([-r,0],X)$ rather than with a point, and it is solved by the method of steps: on each interval of length $r$ the delayed terms are known from the preceding step and the equation is an ordinary initial-value problem, so the Picard–Lindelöf theorem applies step by step and gives a unique solution with a maximal interval of the usual kind. Smoothness of the solution requires compatibility conditions between the history and the equation at the initial time, and is lost at the first step when they fail.

A linear delay equation with constant coefficients has exponential solutions $e^{\lambda t}$ where $\lambda$ solves the transcendental characteristic equation $\lambda = a + be^{-\lambda r}$, with countably many roots and no accumulation in the plane; the solution space is infinite-dimensional, since the datum is a function, and the solution operator $T(t)$ sending a history to its later history is a strongly continuous semigroup, compact for $t \ge r$, with the characteristic roots as its spectrum. The inhomogeneous equation is solved by a variation-of-constants formula with the fundamental solution, computable as an inverse Laplace transform. The zero solution is exponentially stable exactly when all characteristic roots lie in a fixed left half-plane, and for the pure delay this is the condition $ar < \pi/2$, so a lag can destabilise a stable system. For nonlinear equations stability is decided by a Lyapunov functional or by the Razumikhin condition on the history at the instants of maximal $V$.

A functional differential equation $y'(t) = f(t,y_t)$ has the same existence and uniqueness theorem, with the Lipschitz condition taken in the norm of the history space and the right-hand side a function on the Banach space $C([-r,0],X)$; without the Lipschitz condition existence still holds under a complete-continuity hypothesis and uniqueness may fail. For the linear equation the characteristic matrix $\Delta(\lambda) = \lambda I - \int e^{\lambda\theta}d\eta(\theta)$ is entire of exponential type, its zeros form the spectrum, and the solutions split into a finite-dimensional exponential part and a decaying remainder. The general theory of the semigroup $T(t)$, its generator and its spectral decomposition is developed with the evolution equations of this Part.

Summary of Notation

Symbol	Meaning
$X$	Banach state space over $\mathbb{K}$
$r$	Lag, a positive constant unless the lag is variable
$y_t$	History at $t$, $y_t(\theta) = y(t+\theta)$, $\theta \in [-r,0]$
$C = C([-r,0],X)$	History space with the supremum norm
$\phi$	Initial history
$f(t,\phi)$	Right-hand side of the functional equation
$A$, $B$	Coefficient operators of a linear delay equation
$\lambda = a + be^{-\lambda r}$	Characteristic equation of the scalar constant-coefficient delay equation
$\Delta(\lambda)$	Characteristic matrix of a general linear functional equation
$T(t)$	Solution operator on $C$, $T(t)\phi = y_t$
$K(t)$	Fundamental solution of the inhomogeneous linear equation
$V(t,\phi)$	Lyapunov functional; $\dot V$ its right derivative along solutions
$L$	Bounded linear map $C \to X$ of a linear functional equation
$\eta$	Matrix-valued measure representing $L$
Razumikhin condition	A decrease condition imposed when $V$ is at its historical maximum