TR available - Controllability of Recurrent Nets

[Eduardo Sontag]

    COMPLETE CONTROLLABILITY OF CONTINUOUS-TIME RECURRENT NEURAL NETWORKS
               Eduardo D. Sontag and Hector J. Sussmann
            Rutgers Center for Systems and Control (SYCON)
            Department of Mathematics,  Rutgers University

 This paper presents a characterization of controllability for the class of
 (continuous-time) recurrent neural networks.  These are described by
 differential equations of the following type:
 
       x'(t)  =  S [ Ax(t) + Bu(t) ]
 
 where "S" is a diagonal mapping S [a,b,c,...] = (s(a),s(b),s(c),...) and "s" 
 is a scalar real map called the activation function of the network.
 Each coordinate of the vector x(t) is a real-valued variable which represents
 the internal state of a neuron, and each coordinate of u(t) is an external
 input signal applied at time t.  Recurrent networks whose activation s is the
 identity function s(x)=x are precisely the linear systems studied in control
 theory.  With nonlinear s, one obtains general families of recurrent nets.
 
 Controllability means that any state x can be transformed into any other
 possible state z, by means of a suitable input signal u(t) applied on some
 time interval.  When s is the identity (the case typical in control theory),
 controllability can be checked by means of a simple algebraic test due to
 Kalman (1960).  The current paper provides a simple characterization for
 recurrent networks when
 
      s(x) = tanh(x)
 
 is the activation typically used in neural network practice.  The condition is
 very different from the one that applies to linear systems.

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