Inputtostate stability (ISS)^{[1]}^{[2]}^{[3]}^{[4]} is a stability notion widely used to study stability of nonlinear control systems with external inputs. Roughly speaking, a control system is ISS if it is globally asymptotically stable in the absence of external inputs and if its trajectories are bounded by a function of the size of the input for all sufficiently large times. The importance of ISS is due to the fact that the concept has bridged the gap between input–output and statespace methods, widely used within the control systems community. The notion of ISS was introduced by Eduardo Sontag in 1989.^{[5]}
Consider a timeinvariant system of ordinary differential equations of the form

(1) 
where is a Lebesgue measurable essentially bounded external input and is a Lipschitz continuous function w.r.t. the first argument uniformly w.r.t. the second one. This ensures that there exists a unique absolutely continuous solution of the system (1).
To define ISS and related properties, we exploit the following classes of comparison functions. We denote by the set of continuous increasing functions with . The set of unbounded functions we denote by . Also we denote if for all and is continuous and strictly decreasing to zero for all .
System (1) is called globally asymptotically stable at zero (0GAS) if the corresponding system with zero input

(WithoutInputs) 
is globally asymptotically stable, that is there exist so that for all initial values and all times the following estimate is valid for solutions of (WithoutInputs)

(GASEstimate) 
System (1) is called inputtostate stable (ISS) if there exist functions and so that for all initial values , all admissible inputs and all times the following inequality holds

(2) 
The function in the above inequality is called the gain.
Clearly, an ISS system is 0GAS as well as BIBO stable (if we put the output equal to the state of the system). The converse implication is in general not true.
It can be also proved that if , as , then , .
For an understanding of ISS its restatements in terms of other stability properties are of great importance.
System (1) is called globally stable (GS) if there exist such that , and it holds that

(GS) 
System (1) satisfies the asymptotic gain (AG) property if there exists : , it holds that

(AG) 
The following statements are equivalent
1. (1) is ISS
2. (1) is GS and has the AG property
3. (1) is 0GAS and has the AG property
The proof of this result as well as many other characterizations of ISS can be found in the papers ^{[6]} and ^{[7]}
An important tool for the verification of ISS are ISSLyapunov functions.
A smooth function is called an ISSLyapunov function for (1), if , and positive definite function , such that:
and it holds:
The function is called Lyapunov gain.
If a system (1) is without inputs (i.e. ), then the last implication reduces to the condition
which tells us that is a "classic" Lyapunov function.
An important result due to E. Sontag and Y. Wang is that a system (1) is ISS if and only if there exists a smooth ISSLyapunov function for it.^{[7]}
Consider a system
Define a candidate ISSLyapunov function by
Choose a Lyapunov gain by
Then we obtain that for it holds
This shows that is an ISSLyapunov function for a considered system with the Lyapunov gain .
One of the main features of the ISS framework is the possibility to study stability properties of interconnections of inputtostate stable systems.
Consider the system given by

(WholeSys) 
Here , and are Lipschitz continuous in uniformly with respect to the inputs from the th subsystem.
For the th subsystem of (WholeSys) the definition of an ISSLyapunov function can be written as follows.
A smooth function is an ISSLyapunov function (ISSLF) for the th subsystem of (WholeSys), if there exist functions , , , , and a positive definite function , such that:
and it holds
Cascade interconnections are a special type of interconnection, where the dynamics of the th subsystem does not depend on the states of the subsystems . Formally, the cascade interconnection can be written as
If all subsystems of the above system are ISS, then the whole cascade interconnection is also ISS.^{[5]}^{[4]}
In contrast to cascades of ISS systems, the cascade interconnection of 0GAS systems is in general not 0GAS. The following example illustrates this fact. Consider a system given by

(Ex_GAS) 
Both subsystems of this system are 0GAS, but for sufficiently large initial states and for a certain finite time it holds for , i.e. the system (Ex_GAS) exhibits finite escape time, and thus is not 0GAS.
The interconnection structure of subsystems is characterized by the internal Lyapunov gains . The question, whether the interconnection (WholeSys) is ISS, depends on the properties of the gain operator defined by
The following smallgain theorem establishes a sufficient condition for ISS of the interconnection of ISS systems. Let be an ISSLyapunov function for th subsystem of (WholeSys) with corresponding gains , . If the nonlinear smallgain condition

(SGC) 
holds, then the whole interconnection is ISS.^{[8]}^{[9]}
Smallgain condition (SGC) holds iff for each cycle in (that is for all , where ) and for all it holds
The smallgain condition in this form is called also cyclic smallgain condition.
System (1) is called integral inputtostate stable (ISS) if there exist functions and so that for all initial values , all admissible inputs and all times the following inequality holds

(3) 
In contrast to ISS systems, if a system is integral ISS, its trajectories may be unbounded even for bounded inputs. To see this put for all and take . Then the estimate (3) takes the form
and the right hand side grows to infinity as .
As in the ISS framework, Lyapunov methods play a central role in iISS theory.
A smooth function is called an iISSLyapunov function for (1), if , and positive definite function , such that:
and it holds:
An important result due to D. Angeli, E. Sontag and Y. Wang is that system (1) is integral ISS if and only if there exists an iISSLyapunov function for it.
Note that in the formula above is assumed to be only positive definite. It can be easily proved,^{[10]} that if is an iISSLyapunov function with , then is actually an ISSLyapunov function for a system (1).
This shows in particular, that every ISS system is integral ISS. The converse implication is not true, as the following example shows. Consider the system
This system is not ISS, since for large enough inputs the trajectories are unbounded. However, it is integral ISS with an iISSLyapunov function defined by
An important role are also played by local versions of the ISS property. A system (1) is called locally ISS (LISS) if there exist a constant and functions
and so that for all , all admissible inputs and all times it holds that

(4) 
An interesting observation is that 0GAS implies LISS.^{[11]}
Many other related to ISS stability notions have been introduced: incremental ISS, inputtostate dynamical stability (ISDS),^{[12]} inputtostate practical stability (ISpS), inputtooutput stability (IOS)^{[13]} etc.
Consider the timeinvariant timedelay system

(TDS) 
Here is the state of the system (TDS) at time , and satisfies certain assumptions to guarantee existence and uniqueness of solutions of the system (TDS).
System (TDS) is ISS if and only if there exist functions and such that for every , every admissible input and for all , it holds that

(ISSTDS) 
In the ISS theory for timedelay systems two different Lyapunovtype sufficient conditions have been proposed: via ISS LyapunovRazumikhin functions^{[14]} and by ISS LyapunovKrasovskii functionals.^{[15]} For converse Lyapunov theorems for timedelay systems see.^{[16]}
Inputtostate stability of the systems based on timeinvariant ordinary differential equations is a quite developed theory. However, ISS theory of other classes of systems is also being investigated: timevariant ODE systems,^{[17]} hybrid systems.^{[18]}^{[19]} In the last time also certain generalizations of ISS concepts to infinitedimensional systems have been proposed.^{[20]}^{[21]}^{[3]}^{[22]}