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We suggest an approximate method of finding a conformal mapping of an annulusonto an arbitrary bounded doubly connected polygonal domain. The method isbased on the parametric Loewner--Komatu method. We consider smooth oneparameter families $\mathcal{F}(z,t)$ of conformal mappings of concentricannuli onto doubly connected polygonal domains $\mathcal{D}(t)$ which areobtained from a fixed doubly connected polygonal domain $\mathcal{D}$ by makinga finite number of rectilinear slits of variable lengths; meanwhile we do notrequire the family of domains $\mathcal{D}(t)$ to be monotonous. The integralrepresentation for the conformal mappings $\mathcal{F}(z,t)$ has unknown(accessory) parameters. We find a PDE for this family and then deduce a systemof PDEs describing dynamics of accessory parameters and conformal modulus of$\mathcal{D}(t)$ when changing the parameter $t$. We note that the right-handsides of equations in the system of ODEs involve functions standing for speedsof movement of the end points of the slits. This allows us to controlcompletely dynamics of the slits and to seek their agreed change, if we havemore than one slit. We also give some results of numerical calculations whichshow the efficiency of the suggested method.

One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains