How can I define my controller using TypeScript?

There are 2 different ways to tackle this:

• still using $scope
• using controllerAs (recommended)

using $scope

class CustomCtrl{
    static $inject = ['$scope', '$http', '$templateCache'];
    constructor (
            private $scope,
            private $http,
            private $templateCache
    ){
        $scope.search = this.search;
    }

    private search (search) {
        debugger;
        var Search = {
            AccountId: search.AccountId,
            checkActiveOnly: search.checkActiveOnly,
            checkParentsOnly: search.checkParentsOnly,
            listCustomerType: search.listCustomerType
        };
        this.$scope.customer = [];
        this.$scope.ticket = [];
        this.$scope.services = [];
        this.$http.put('<%=ResolveUrl("API/Search/PutDoSearch")%>', Search).
                success((data, status, headers, config) => {
                    debugger;
                    this.$scope.cust_File = data[0].customers;
                    this.$scope.ticket_file = data[0].tickets;
                    this.$scope.service_file = data[0].services;
                }).
                error((data, status) => {
                    console.log("Request Failed");
                });

    }
}

Using controllerAs

class CustomCtrl{
    public customer;
    public ticket;
    public services;
    public cust_File;
    public ticket_file;
    public service_file;

    static $inject = ['$scope', '$http', '$templateCache'];
    constructor (
            private $http,
            private $templateCache
    ){}

    private search (search) {
        debugger;
        var Search = {
            AccountId: search.AccountId,
            checkActiveOnly: search.checkActiveOnly,
            checkParentsOnly: search.checkParentsOnly,
            listCustomerType: search.listCustomerType
        };
        this.customer = [];
        this.ticket = [];
        this.services = [];
        this.$http.put('<%=ResolveUrl("API/Search/PutDoSearch")%>', Search).
                success((data, status, headers, config) => {
                    debugger;
                    this.cust_File = data[0].customers;
                    this.ticket_file = data[0].tickets;
                    this.service_file = data[0].services;
                }).
                error((data, status) => {
                    console.log("Request Failed");
                });

    }
}

If you switch from $scope to controllerAs your view would change from:

<div ng-controller="CustomCtrl">
  <span>{{customer}}</span>
</div>

to:

<div ng-controller="CustomCtrl as custom">
  <span>{{custom.customer}}</span>
</div>

where custom is a representation of the controller so you are explicitly telling what you are binding to in your markup.

Note
$inject is a way to provide angular with information about what dependencies to inject into your controller at run time even when the code has been minified (strings don’t get minified)