## 2 Methods
### Determination of a Transfer Matrix Used to Determine the Objective Function in Optimization Issues
The structure's load Fourier transform is expressed by the formula:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]5  
If we determine the Fourier retransform of the structure's excitation forces vector, we get the following relationship:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]6  
The equation for \(\mathbf{q}(t)\) determination based on the Duhamel integral is often used in the literature. The matrix form of the equation is expressed by the formula:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]7  
where \(\mathbf{q}(t)\) is a response vector, \(\mathbf{h}(t)\) is a matrix of impulse response, and \(\mathbf{p}(t)\) is the vector of excitation forces. Since it was assumed that the random excitation is a stationary process, the dynamic response is also a stationary process. Due to this fact, the correlation matrix of dynamic response of structure can be written as:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]8  
Inserting the Equation (47) into the Equation (48), we assume:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]9If we assume that the correlation of excitation forces is:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]0  
we can write the formula (49) in the form:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]1  
where \(\mathbf{H}^{\mathrm{H}}(j\omega)\) denotes the complex conjugate and transpose of frequency response matrix \(\mathbf{H}(j\omega)\) (Hermitian matrix) and \(\mathbf{S}_{p}(j\omega)\) is a full matrix of spectral density of random excitation. If we determine the values of correlation matrix at the moment \(\tau=0\), the dependency for a matrix of variance of a dynamic response of a system is obtained:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]2  
where \(\mathbf{S}_{q}(j\omega)\) is a matrix of double-sided spectral density of responses \(\mathbf{S}_{q}(j\omega)=\mathbf{H}(j\omega)\mathbf{S}_{p}(j\omega)\mathbf{H} ^{\mathrm{H}}(j\omega)\). In the case of acting on the system of excitation of white noise nature \(\mathbf{S}_{p}(\omega)=\mathbf{I}\), the matrix of spectral densities of response is the product of frequency response matrix \(\mathbf{H}(j\omega)\) and a matrix conjugated and transposed to \(\mathbf{H}^{\mathrm{H}}(j\omega)\). The standard deviation of response can be expressed by the formula:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]3  
Now consider the \(p\)-norm of matrix \(\mathbf{A}=\big{[}A_{ij}\big{]}\). Let \(p\geq 1\) be a real number. The \(p\)-norm (also called \(\mathcal{L}_{p}\)-norm) is expressed by the formula:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]4  
For \(p=2\), we get the \(\mathcal{L}_{2}\)-norm:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]5  
For a multivariable (multiple inputs and multiple outputs (MIMO)), LTI system with transfer matrix \(\mathbf{H}(s)=\big{[}H_{ij}(s)\big{]}\), the definition (55) generalizes to \(\mathfrak{H}_{2}\)-norm:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]6  
The notation \(\mathfrak{H}_{2}\) (instead of \(\mathcal{L}_{2}\)) is due to the fact that the function spaces that, in addition to having finite \(\mathcal{L}_{p}\) norms on the imaginary axis, are bounded and analytic functions in the right-half plane (RHP) (i.e., with no poles in the RHP) are called Hardy spaces \(\mathfrak{H}_{p}\). Thus, stable transfer functions belong to these spaces, provided the associated integral is finite. So we can see that the \(\mathfrak{H}_{2}\)-norm of the transfer matrix \(\mathbf{H}(s)\) is the sum of diagonal values of the standard deviations \(\mathbf{\sigma}_{q}\) matrix for white noise excitation obtained from the formula (53).