This paper is the first part of a study on the suboptimal full information \(H^\infty\) problem for a well-posed linear system with input space \(U\), state space \(H\), and output space \(Y\). If \(\omega\in\mathbb{R}\), a (causal) \(\omega\)-stable well-posed linear system on \((U,H,Y)\) is a quadruple \(\Psi = \left[\begin{smallmatrix} {\mathcal A} & {\mathcal B}\\{\mathcal C} & {\mathcal D}\end{smallmatrix}\right]\), where \({\mathcal A}, {\mathcal B},{\mathcal C}\) and \(\mathcal D\) are bounded linear operators of the following type: (i) \({\mathcal A}(t):H\to H\) is a strongly continuous semigroup of bounded linear operators on \(H\) satisfying \(\sup_{t\in\mathbb{R}^+}\|e^{-\omega t}{\mathcal A}(t)\|< \infty\); (ii) \({\mathcal B} : L^2_\omega(\mathbb{R},U)\to H\) satisfies \({\mathcal A}(t){\mathcal B}u={\mathcal B}\tau(t)\pi_u\) for all \(x\in H\) and all \(t\in\mathbb{R}^+\); (iii) \({\mathcal C} : H\to L^2_\omega(\mathbb{R};Y)\) satisfies \({\mathcal C}{\mathcal A}(t)x = \pi_+\tau(t){\mathcal C}x\) for all \(x\in H\) and all \(t\in \mathbb{R}^+\); (iv) \({\mathcal D}:L^2_\omega(\mathbb{R},U)\to L_\omega^2(\mathbb{R};Y)\) satisfies \(\tau(t){\mathcal D}u = {\mathcal D}\tau(t) u\), \(\pi_-{\mathcal D}\pi_+u= 0\) and \(\pi_+{\mathcal D}\pi_-u={\mathcal C}{\mathcal B}u\) for all \(u\in L^2_\omega(\mathbb{R};U)\) and all \(t\in\mathbb{R}\). To a system \(\Psi\) is associated a cost function \[ Q(x_0,u)=\int_{\mathbb{R}^+}\langle y(s),Jy(s)\rangle_Y ds \] where \(y\in L^2_{\text{loc}}(\mathbb{R}^+; Y)\) is the output of the system with initial state \(x_0\in H\) and control \(u\in L^2_{\text{loc}}(\mathbb{R}^+;U)\), and \(J\) is a self-adjoint operator on \(Y\). The cost function \(Q\) is quadratic in \(x_0\) and \(u\), and (in the stable case) it is assumed that the second derivative of \(Q(x_0, u)\) with respect to \(u\) is non-singular. This implies that, for each \(x_0 \in H\), there is a unique critical control \(u^{\text{crit}}\) such that the derivative of \(Q(x_0,u)\) with respect to \(u\) vanishes at \(u = u^{\text{crit}}\). It is shown that \(u^{\text{crit}}\) can be written in feedback form whenever the input-output map of the system has a coprime factorization with a \((J, S)\)-inner numerator, here \(S\) is a particular self-adjoint operator on \(U\). A number of properties of this feedback representation are established, such as the equivalence of the \((J, S)\)-losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace.